Probability and Statistical Physics Seminar
Probability and Statistical Physics Seminar
Title: Grothendeick \(L_p\) problem for Guaussian matrices
Abstract: The Grothendieck \(L_p\) problem is defined as an optimization problem that maximizes the quadratic form of a Gaussian matrix over the unit \(L_p\) ball. The \(p=2\) case corresponds to the top eigenvalue of the Gaussian Orthogonal Ensemble, while \(p=\infty\) this problem is known as the ground state energy of the Sherrington-Kirkpatrick mean-field spin glass model and its limit can be expressed by the famous Parisi formula. In this talk, I will describe the limit of this optimization problem for general \(p\) and discuss some results on the behavior of the near optimizers along with some open problems. This is based on a joint work with Arnab Sen.
Title: Random planar geometry and the directed landscape
Abstract: Consider the lattice \(\mathbb Z^2\), and assign length 1 or 2 to every edge by flipping a series of independent fair coins. This gives a random weighted graph, and looking at distances in this graph gives a random planar metric. This model, along with most natural models of random planar metrics and random interface growth (the so-called `KPZ universality class'), is expected to converge to a universal scaling limit: the directed landscape. The goal of this talk is to introduce this object, describe some of its properties, and describe at least one model where we can actually prove convergence.
Title: Geodesic networks in the directed landscape
Abstract: The directed landscape is a random directed metric on the plane that is the scaling limit for models in the KPZ universality class. In this metric, typical pairs of points are connected by a unique geodesic. However, certain exceptional pairs are connected by more exotic geodesic networks. The goal of this talk is to describe a full classification for these exceptional pairs. I will also discuss some connections with other models of random geometry.
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Title: Recent progress on random graph matching problems
Abstract: A basic goal for random graph matching is to recover the vertex correspondence between two correlated graphs
from an observation of these two unlabeled graphs. Random graph matching is an important and active topic in
combinatorial statistics: on the one hand, it arises from various applied fields such as social network analysis,
computer vision, computational biology and natural language processing; on the other hand, there is also a deep
and rich theory that is of interest to researchers in statistics, probability, combinatorics, optimization, algorithms
and complexity theory.
Recently, extensive efforts have been devoted to the study for matching two correlated Erdős–Rényi graphs, which
is arguably the most classic model for graph matching. In this talk, we will review some recent progress on this
front, with emphasis on the intriguing phenomenon on (the presumed) information-computation gap. In particular,
we will discuss progress on efficient algorithms thanks to the collective efforts from the community. We will also
point out some important future directions, including developing robust algorithms that rely on minimal
assumptions on graph models and developing efficient algorithms for more realistic random graph models.
This is based on joint works with Hang Du, Shuyang Gong, Zhangsong Li, Zongming Ma, Yihong Wu and
Jiaming Xu.
Title: Conformal removability of non-simple Schramm-Loewner Evolutions
Abstract: We consider the Schramm-Loewner evolution (SLE_{kappa}) for kappa in (4,8), which is the regime that the curve is self-intersecting but not space-filling. We let K be the set of kappa in (4,8) for which the adjacency graph of connected components of the complement of an SLE_{kappa} is almost surely connected, meaning that for every pair of complementary components U,V there exist complementary components U_1,...,U_n with U_1 = U, U_n = V, and the boundaries of U_i and U_{i+1} have non-empty intersection for every i between 1 and n-1. We show that the range of an SLE_{kappa} for kappa in K is almost surely conformally removable, which answers a question of Sheffield. As a step in the proof, we construct the canonical conformally covariant volume measure on the cut points of an SLE_{kappa} for kappa in (4,8) and establish a precise upper bound on the measure that it assigns to any Borel set in terms of its diameter.
Title: Boundary current fluctuations for the half space ASEP
Abstract:
Abstract: The half space asymmetric simple exclusion process (ASEP) is an interacting particle system on the half line, with particles allowed to enter/exit at the boundary. I will discuss recent work on understanding fluctuations for the number of particles in the half space ASEP started with no particles, which exhibits the Baik-Rains phase transition between GSE, GOE, and Gaussian fluctuations as the boundary rates vary. As part of the proof, we find new distributional identities relating this system to two other models, the half space Hall-Littlewood process, and the free boundary Schur process, which allows exact formulas to be computed.
Title: Tail bounds for the averaged empirical distribution on a geodesic in first-passage percolation
Abstract: We establish tail estimates for the averaged empirical distribution of edge weights along geodesics in first-passage percolation on Z^d. Our upper bounds also come with exponentially decaying tail estimates for the empirical distribution. As an application, we show that even for an edge weight distribution with no positive moment, the limit point of the averaged empirical distribution of finite geodesics could have finite positive moments of all orders. (Joint work with Michael Damron, Christopher Janjigian, Jack Hanson and Wai-Kit Lam.)
Title: Conformal welding of Liouviile quantum gravity triangles
Abstract: Liouville quantum gravity (LQG) is a natural model describing random surfaces. A powerful tool of studying LQG is the conformal welding, where we glue two LQG surfaces together into a single LQG surface with the interface being Schramm-Loewner evolution (SLE) type curves. We will present some recent results on the conformal welding of LQG surfaces with three marked points, along with its extensions and applications, including connections with SLE reversibility, multiple SLE and conformal loop ensemble (CLE) boundary touching probability. Based on joint works with Ang and Sun.
2:30-3:30pm Ben McKenna - Harvard University
Title: Random determinants, the elastic manifold, and landscape complexity beyond invariance
Abstract: The Kac-Rice formula allows one to study the complexity of high-dimensional Gaussian random functions (meaning asymptotic counts of critical points) via the determinants of large random matrices. We present new results on determinant asymptotics for non-invariant random matrices, and use them to compute the (annealed) complexity for several types of landscapes. We focus especially on the elastic manifold, a classical disordered elastic system studied for example by Fisher (1986) in fixed dimension and by Mézard and Parisi (1992) in the high-dimensional limit. We confirm recent formulas of Fyodorov and Le Doussal (2020) on the model in the Mézard-Parisi setting, identifying the boundary between simple and glassy phases. Joint work with Gérard Ben Arous and Paul Bourgade.
3:30-4:30pm Sayan Das - Columbia University
Title: Geometry of half-space log-gamma polymers
Abstract: The half-space directed polymer is a variant of directed polymer which studies how polymers behave in presence of an attractive wall. Depending on the strength of the boundary, the polymers are expected to have two distinct phases: the bound phase and the unbound phase. In this talk, I will focus on the half-space polymer model with log-gamma weights which makes the model integrable. I will describe our results in the unbound phase where we obtain KPZ exponents and in the bound phase where we obtain stochastic boundedness of the endpoint. Our proof proceeds by constructing the half-space log-gamma line ensemble which has a novel feature of attraction/repulsion at the boundaries. Based on two joint works: one with Guillaume Barraquand and Ivan Corwin, and one with Weitao Zhu.
Title: Scaling limit of soliton lengths in a multicolor box-ball system
Abstract: The box-ball systems are integrable cellular automata whose long-time behavior is characterized by soliton solutions, with rich connections to other integrable systems such as the Korteweg-de Vries equation. In this paper, we consider the multicolor box-ball system with two types of random initial configurations and obtain sharp scaling limits of the soliton lengths as the system size tends to infinity. We obtain sharp scaling of soliton lengths in the multicolor case, which turns out to be different from the single color case as established in Levine, Lyu, Pike 2017. A large part of our analysis is devoted to study the associated carrier process, which is a multi-dimensional Markov chain on the orthant, whose excursions and running maxima are closely related to soliton lengths. We establish the sharp scaling of its ruin probabilities, Skorokhod decomposition, strong law of large numbers, and weak diffusive scaling limit to a semimartingale reflecting Brownian motion with explicit parameters. We also establish and utilize complementary descriptions of the soliton lengths and numbers in terms of the modified Greene-Kleitman invariants for the box-ball systems and associated circular exclusion processes.
Based on a joint work with Joel Lewis, Pasha Pylyavskyy, and Arnay Sen.
Title: Random surface, Planar Lattice Model, and Conformal Field Theory
Abstract: Liouville quantum gravity (LQG) is a theory of random surfaces that originated from string theory. Schramm Loewner evolution (SLE) is a family of random planar curves describing scaling limits of many 2D lattice models at their criticality. Before the rigorous study via LQG and SLE in probability, random surfaces and scaling limits of lattice models have been studied via another approach in theoretical physics called conformal field theory (CFT) since the 1980s. In this talk, I will demonstrate how a combination of ideas from LQG/SLE and CFT can be used to rigorously prove several long standing predictions in physics on random surfaces and planar lattice models, including the law of the random modulus of the scaling limit of uniform triangulation of the annular topology, and the crossing formula for critical planar percolation on an annulus. I will then present some conjectures which further illustrate the deep and rich interaction between LQG/SLE and CFT.
Based on joint works with Ang, Holden, Remy, Xu, and Zhuang.
Title: Stochastic waves on metric graphs and their genealogies
Abstract:
Stochastic reaction-diffusion equations are important models in mathematics and in applied sciences such as spatial population genetics and ecology. These equations describe a quantity (density/concentration of an entity) that evolves over space and time, taking into account random fluctuations.
However, for many reaction terms and noises, the solution notion of these equations is still missing in dimension two or above, hindering the study of spatial effect on stochastic dynamics through these equations.
In this talk, I will discuss a new approach, namely, to study these equations on general metric graphs that flexibly parametrize the underlying space. This enables us to not only bypass the ill-posedness issue of these equations in higher dimensions, but also assess the impact of space and stochasticity on the coexistence and the genealogies of interacting populations. We will focus on the computation of the probability of extinction, the quasi-stationary distribution, the asymptotic speed and other long-time behaviors for stochastic reaction-diffusion equations of Fisher-KPP type.
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Title: Voronoi Tessellations without Nuclei
Abstract: Given a discrete set of points in a metric space, called nuclei, one associates to each such nucleus its Voronoi cell, which consists of all points closer to it than to other nuclei. This construction is widely used in mathematics, science, and engineering; it is even used in baking. In Euclidean space, one commonly uses a homogeneous Poisson point process to assign the locations of the nuclei. As the intensity of the point process tends to 0, the nuclei spread out and disappear in the limit, with each pair of points eventually belonging to the same cell. Surprisingly, this does not happen in other settings such as hyperbolic space; instead, one obtains a Voronoi tessellation without nuclei! We describe properties of such a limiting tessellation, as well as analogous behavior on Cayley graphs of finitely generated groups. We will illustrate results with many pictures and several animations. The talk is based on work of Sandeep Bhupatiraju and joint work in progress with Matteo d'Achille, Nicolas Curien, Nathanael Enriquez, and Meltem Unel. We will not assume knowledge of Poisson point processes or of hyperbolic space.
Title: Monotonicity for Continuous-Time Random Walks
Abstract: Variable-speed, continuous-time random walk on a graph is given by an assignment of nonnegative rates to its edges. There are independent Poisson processes associated to the edges with the given rates. When a walker is at a vertex, it jumps to a neighbor at the time of the next event that occurs for the corresponding incident edges. In the case of a Cayley graph of a finitely generated group, we are particularly interested in the setting where the edge rates depend only on the corresponding generators. Our lecture is concerned with monotonicity in the rates for various fundamental properties of random walks. We will survey results, counterexamples, and open questions. We will give general ideas of proofs, but avoid technicalities.
Most of the talk will be devoted to two questions on Cayley graphs: On infinite graphs, we ask about the limiting linear rate of escape, i.e., the limit of the distance divided by the time. Does this increase when the rates are increased? On finite graphs, we ask about the convergence to the stationary (uniform) distribution. Does this happen faster when the rates are increased? It turns out that both questions have surprising answers. This is joint work with Graham White.
Title: The stationary horizon as a universal object for KPZ models
Abstract: Abstract: The last 5-10 years has seen remarkable progress in constructing the central
objects of the KPZ universality class, namely the KPZ fixed point and directed landscape.
In this talk, I will discuss a third central object known as the stationary horizon (SH).
The SH is a coupling of Brownian motions with drifts, indexed by the real line, and it
describes the unique coupled invariant measures for the directed landscape.
I will talk about how the SH appears as the scaling limit of several models,
including Busemann processes in last-passage percolation and the TASEP speed process.
I will also discuss how the SH helps to describe the collection of infinite geodesics
in all directions for the directed landscape.
Based on joint work with Timo Seppäläinen and Ofer Busani.
Title: Singularity and the least singular value of random symmetric matrices
Abstract: Consider the random symmetric \(n\times n\) matrix whose entries on and above the diagonal are independent and uniformly chosen from \(\{-1,1\}\). How often is this matrix singular? A moment's thought reveals that if two rows or columns are equal then the matrix is singular, so the singularity probability is bounded below by \(2^{-n(1 + o(1))}\). Proving any sort of upper bound on the singularity probability turns out to be difficult, with results coming slowly over the past two decades. I'll discuss work which shows the first exponential upper bound on this probability as well as extensions to more quantitative notions of invertibility. Along the way, I'll describe some tools---both old and new---which are powerful and (hopefully) interesting in their own right. This talk is based on joint work with Marcelo Campos, Matthew Jenssen, and Julian Sahasrabudhe.
Title: Rewriting History in Integrable Stochastic Particle Systems
Abstract:
Imagine two cars, slow (S) and fast (F), driving to the right on a discrete 1-dimensional lattice according to some random walk mechanism, and such that the cars cannot pass each other. We consider two systems, SF and FS, depending on which car is ahead. It is known for some time (through connections to symmetric functions and the RSK correspondence) that if at time 0 the cars are immediate neighbors, the trajectory of the car that is behind is the same (in distribution) in both systems. However, this fact fails when the initial locations of the cars are not immediate neighbors. I will explain how to recover the identity in distribution by suitably randomizing the initial condition in one of the systems.
This result arises in our recent work on multiparameter stochastic systems (where the parameters are speeds attached to each car) in which the presence of parameters preserves the quantum integrability. This includes TASEP (totally asymmetric simple exclusion process), its deformations, and stochastic vertex models, which are all integrable through the Yang-Baxter equation (YBE). In the context of car dynamics, we interpret YBEs as Markov operators intertwining the transition semigroups of the dynamics of the processes differing by a parameter swap. We also construct Markov processes on trajectories which "rewrite the history" of the car dynamics, that is, produce an explicit monotone coupling between the trajectories of the systems differing by a parameter swap.
Title: Dimer Model Fluctuations via Perfect t-embeddings
Abstract: Chelkak, Laslier, and Russkikh introduce a new type of graph embedding called a t-embedding, and use it to prove the convergence of dimer model height fluctuations to a Gaussian Free Field (GFF) in a naturally associated metric, under certain technical assumptions. Building on a work of Chelkak and Ramassamy, we study the properties of t-embeddings of uniform Aztec diamond graphs, and in particular utilize the integrability of the “shuffling algorithm” on these graphs to provide a precise asymptotic analysis of t-embeddings, and in particular verify the validity of the technical assumptions required for convergence. As a consequence, we complete a new proof of GFF fluctuations for the dimer model height function on the uniformly weighted Aztec diamond.
Title: Cutoff in the Glauber dynamics for the Gaussian free field
Abstract: The Gaussian free field (GFF) is a canonical model of random surfaces, generalizing the Brownian bridge to two dimensions. It arises naturally as the stationary solution to the stochastic heat equation with additive noise (SHE), and together the SHE and GFF are expected to be the universal scaling limit of many random surface evolutions arising in lattice statistical physics. We study the mixing time (time to converge to stationarity, when started out of equilibrium) for the central pre-limiting object, the discrete Gaussian free field (DGFF) evolving under the Glauber dynamics. In joint work with S. Ganguly, we establish that on a box of side-length $n$ in $\mathbb Z^2$, the Glauber dynamics for the DGFF exhibits the cutoff phenomenon, mixing exactly at time $\frac{2}{\pi^2} n^2 \log n$.
Title: Colored noise and parabolic Anderson model on Torus
Abstract: We construct an intrinsic family of Gaussian noises on compact Riemannian manifolds which we call the colored noise on manifolds. It consists of noises with a wide range of singularities. This family of noises allows the study of the parabolic Anderson model (PAM) on compact manifolds in high dimensions in the Ito sense. We started our investigation on a toy model, an n-dimensional torus, and established existence and uniqueness of the solution, as well as some sharp bounds on the second moment of the solution. It sheds a light to the study of PAM on more general manifolds.
Title: Mobility Edge for Lévy Matrices
Abstract: Lévy matrices are symmetric random matrices whose entries are in the domain of attraction of an \(\alpha\) stable law. For \(\alpha < 1\), it had been predicted that these matrices exhibit an Anderson transition, also called a mobility edge, a point in the spectrum where eigenvector behavior sharply transitions from delocalized to localized. In this talk, we describe recent results that establish the existence and also explicitly compute the location of this mobility edge for Lévy matrices. This is based on joint work with Charles Bordenave and Patrick Lopatto.
Title: Some recent progress in the weak noise theory of the KPZ equation Abstract:
Abstract: The Kardar-Parisi-Zhang (KPZ) equation is a nice model for random interface growth. In this talk, we will study the Freidlin–Wentzell LDP for the KPZ equation using the variational principle. Such an approach goes under the name of the weak noise theory in physics. We will explain how to extract various limits of the most probable shape of the KPZ equation in the setting of the Freidlin–Wentzell LDP. The talk is based on several joint work with Pierre Yves Gaudreau Lamarre and Li-Cheng Tsai.
Title: Ergodicity and synchronization of the KPZ equation
Abstract: The Kardar-Parisi-Zhang (KPZ) equation on the real line is well-known to have stationary distributions modulo additive constants given by Brownian motion with drift. In this talk, we will discuss some results-in-progress which show that these distributions are totally ergodic and present some progress toward the conjecture that these are the only ergodic stationary distributions of the KPZ equation. The talk will discuss our coupling of Hopf-Cole solutions, which enables us to study the KPZ equation started from any measurable function valued initial condition. Through this coupling, we give a sharp characterization of when such solutions explode, show that all non-explosive functions become instantaneously continuous, and then study the problem of ergodicity on a natural topology on the space of non-explosive continuous functions (mod constants) in which the equation defines a Feller process. We show that any ergodic stationary distribution on this space is either a Brownian motion with drift or a process of a very peculiar form which will be described in the talk.
Based on joint works with Tom Alberts, Firas Rassoul-Agha, and Timo Seppäläinen.
Title: Coalescence of geodesics and the BKS midpoint problem in planar first-passage percolation
Abstract: First-passage percolation studies the geometry obtained from a random perturbation of Euclidean geometry. In the discrete planar setting, one assigns random, independent and identically distributed, lengths to the edges of the lattice Z^2 and studies the resulting geodesics - paths of minimal length between points. While the physics literature presents an elaborate picture for the behavior of the model, placing it in the KPZ universality class, mathematical progress remains rather limited. Aside from the random geometry perspective, the model also enjoys close ties with disordered spin systems.
The talk will give an introduction to planar first-passage percolation, followed by a description of recent progress, joint with Barbara Dembin and Dor Elboim, on the problem of coalescence of geodesics. Our result shows, under mild assumptions, that geodesics with nearby starting and ending points have significant overlap, coalescing on all but small portions near their endpoints. The statement is quantified, with power-law dependence of the involved quantities on the length of the geodesics. The result further implies the first quantitative progress on a problem of Benjamini-Kalai-Schramm (2002), which asks to prove that geodesics typically do not pass at the midpoint of the straight line segment connecting their endpoints.
No prior knowledge of first-passage percolation will be assumed.
Title: Large genus geometry of triangulated surfaces
Abstract: A triangulated surface is a compact surface formed by gluing together unit equilateral triangles. It admits a conformal structure coming from the conformal structure on each triangle. When the genus is at least two, it is also a hyperbolic surface in a natural way. Brooks and Makover proved results about the geometric properties of random triangulated surfaces in the large genus limit. Mirzakhani proved similar results for random hyperbolic surfaces. These results, along with many others, suggest that the large genus geometry of random triangulated surfaces mirrors the large genus geometry of random hyperbolic surfaces. In this talk, I will explain the underlying reasons for this similarity.
Title: Superdiffusivity and localization of continuous polymers
Abstract: We study the Continuum Directed Random Polymer (CDRP) which arises as a universal scaling limit of discrete directed polymers. In this talk, I will present some of the recent progress in understanding the geometry of the CDRP. In particular, I will show CDRP exhibits pointwise localization and pathwise tightness under 2/3 scaling, confirming the existing predictions in polymer literature under continuous setting. I will explain how our results also shed light on certain properties of the KPZ equation (free energy of the CDRP) such as ergodicity and limiting Bessel behaviors around the maximum. This is based on two joint works with Weitao Zhu.
Title: Talagrand’s selector process conjecture and suprema of positive empirical processes
Abstract: Understanding suprema of stochastic processes is an important subject in probability theory with many applications. While much is known in the case of Gaussian processes thanks to Talagrand’s celebrated majorizing measure theorem, moving beyond the Gaussian case is a much more challenging quest. In this talk, I will discuss recent joint work with Jinyoung Park that resolves a conjecture of Talagrand on extreme events of suprema of certain stochastic processes driven by sparse Bernoulli random variables (known as selector processes), and a question of Talagrand on general positive empirical processes. Combining with the recent resolution of the (generalized) Bernoulli conjecture, this gives the first steps towards the last missing piece in the study of suprema of general empirical processes. The proof of Talagrand’s conjecture is combinatorial and easily leads to the proof of the Kahn-Kalai conjecture, an important question in probabilistic combinatorics and random graph theory.
Title: Cointegration, S&P, and random matrices
Abstract:
Cointegration is a property of an N-dimensional time series, which says that
each individual component is non-stationary (growing like a random walk),
but there exists a stationary linear combination. Testing procedures for the
presence of cointegration has been extensively studied in statistics and
economics, but most results are restricted to the case when N is much
smaller than the length of the time series. I will discuss the recently
discovered mathematical structures, which make the large N case
accessible.
On the applied side we will see a remarkable match between predictions of
random matrix theory and behavior of S&P 100 stocks. On the theoretical
side we will see how ideas from statistical mechanics and asymptotic
representation theory play a crucial role in the analysis.
At 2:30 pm: Yair Shenfeld - MIT
Title: The Langevin transport map and renormalization groups
Abstract: A basic problem in probability theory and engineering is to find a way of representing a complex probability measure as a simpler probability measure under some transformation. A desirable property of such a transformation is its Lipschitz property, since it allows for information from the simpler probability measure to be transferred to the complex measure. While various transformations (optimal transport, Knothe-Rosenblatt rearrangement) exist, establishing their regularity is a difficult problem. In this talk, I will discuss the Lipschitz properties of the Langevin transport map which is constructed infinitesimally along the Langevin dynamics. I will show that this map is Lipschitz in many settings where no other Lipschitz transport maps are known to exist. I will conclude the talk by introducing a new connection between the Langevin transport map and renormalization groups methods from quantum and statistical field theories.
At 3:30 pm: Vlad Mărgărint - University of Colorado Boulder
Title: SLE theory, Rough Paths and beyond
Abstract: In this talk, I will give an overview of some work carried out at the intersection of Rough Path Theory and Schramm-Loewner Evolutions (SLE) Theory. Specifically, I will cover a study of the Loewner Differential Equation using Rough Path techniques (and beyond). The Loewner Differential Equation describes the evolution of a family of conformal maps. We rephrase this in terms of Singular Rough Differential Equations. In this context, it is natural to study questions on the stability, and approximations of solutions of this equation. First, I will give an introduction to these two theories, and then I will present a result on the continuity of the dynamics and related objects in a natural parameter that appears in the problem. The first approach will be based on Rough Path Theory, and a second approach will be based on a constructive method of independent interest: the square-root interpolation of the Brownian driver of the Loewner Differential Equation. This is based on joint work with Dmitry Belyaev and Terry Lyons. If time permits, I will also touch on some further applications to the simulations of the SLE traces, that is based on joint works with James Foster and Jiaming Chen.
Title: On the TAP equations and the local magnetization of the Sherrington-Kirkpatrick (SK) model
Abstract: The TAP equations for the Sherrington-Kirkpatrick model are a set of high-dimensional, nonlinear, fixed-point equations of the local magnetization. In the seminal work [Comm. Math. Phys., 325(1):333-366, 2014], Bolthausen introduced an iterative scheme that produces an asymptotic solution to the TAP equations if the model lies below the Almeida-Thouless transition line (“high temperature regime”). However, it was unclear if this asymptotic solution coincides with the local magnetization. In this talk, I will introduce a new iterative scheme, motivated by the cavity equations of the SK model, and show that the new scheme is asymptotically the same as the so-called Approximate Message Passing (AMP) algorithm, a generalization of Bolthausen's iteration, that has been popularly adapted in compressed sensing, Bayesian inferences, etc. Based on this, we confirm that our cavity iteration (and hence Bolthausen's iteration) converges to the local magnetization as long as the overlap is locally uniformly concentrated. If time permits, I will also briefly discuss the TAP equations in the low temperature regime. The talk is based on joint works with Wei-Kuo Chen (University of Minnesota).
Title: Understanding the upper tail behaviour of the KPZ equation via the tangent method
Abstract: The Kardar-Parisi-Zhang (KPZ) equation is a canonical non-linear stochastic PDE believed to describe the evolution of a large number of planar stochastic growth models which make up the KPZ universality class. A particularly important observable is the one-point distribution of its analogue of the fundamental solution, which has featured in much of its recent study. However, in spite of significant recent progress relying on explicit formulas, a sharp understanding of its upper tail behaviour has remained out of reach. In this talk we will discuss a geometric approach, closely connected to the tangent method introduced by Colomo-Sportiello and rigorously implemented by Aggarwal for the six-vertex model. The approach utilizes a Gibbs resampling property of the KPZ equation and yields a sharp understanding for a large class of initial data.
Joint work with Shirshendu Ganguly.
Title: Almost-optimal regularity conditions in the CLT for Wigner matrices
Abstract: We consider linear spectral statistics for test functions of low
regularity and Wigner matrices with smooth entry distribution. We show that for functions
in the Sobolev space H1/2+ε or the space C1/2+ε that are supported
within the spectral bulk of the semicircle distribution, the variance remains bounded
asymptotically. As a consequence, these linear spectral statistics have asymptotic Gaussian
fluctuations with the same variance as in the CLT for functions of higher regularity, for any ε>0.
This result is nearly optimal in the sense that the linear statistic are known not to remain bounded
for functions in H1/2, and was previously known only for matrices in Gaussian Unitary Ensemble.
Title: Localization and Delocalization in Erdős–Rényi graphs
Abstract: We consider the Erdős–Rényi graph on N vertices with edge probability d/N. It is well
known that the structure of this graph changes drastically when d is of order log N. Below this threshold
it develops inhomogeneities which lead to the emergence of localized eigenvectors, while the majority of the
eigenvectors remains delocalized. In this talk, I will present the phase diagram depicting these localized
and delocalized phases and our recent progress in establishing it rigorously.
This is based on joint works with Raphael Ducatez and Antti Knowles.
Title: One force--one solution and semi-infinite polymers for KPZ on the line
Abstract: In this talk, I will present recent results on the uniqueness, ergodicity, and attractiveness of stationary solutions to the Kardar-Parisi-Zhang (KPZ) equation on the real line. It is known that this equation admits Brownian motion with a linear drift as a stationary solution. We show that these solutions are attractive, a principle known as one force--one solution (1F1S): the solution to the KPZ equation started in the distant past from an initial condition with a given velocity will converge almost surely to a Brownian motion with that same drift. As a result, we deduce that these stationary measures are ergodic and that there are no other ergodic measures. Furthermore, we can couple all these stationary solutions so that the above attractiveness holds simultaneously (i.e. on a single full measure event) for all but countably many (random) asymptotic velocities. I will also discuss how these ideas connect to the continuum directed polymer and in particular to semi-infinite continuum polymer measures, as well as the structure of the random set of exceptional velocities on which 1F1S fails. This is joint work with Tom Alberts, Chris Janjigian, and Timo Seppalainen.
Title: The limit shape of the Leaky Abelian Sandpile Model
Abstract: The leaky abelian sandpile model (Leaky-ASM) is a growth model in which n grains of sand start at the origin in the square lattice and diffuse according to a toppling rule. A site can topple if its amount of sand is above a threshold. In each topple a site sends some sand to each neighbor and leaks a portion 1-1/d of its sand. This is a dissipative generalization of the Abelian Sandpile Model, which corresponds to the case d=1.
We will discuss how, by connecting the model to a certain killed random walk on the square lattice, for any fixed d>1, an explicit limit shape can be computed for the region visited by the sandpile when it stabilizes.
We will also discuss the limit shape in the regime when the dissipation parameter d converges to 1 as n grows, as this is related to the ordinary ASM with a modified initial configuration.
Title: Fractal Dimension in the Directed Landscape in the Temporal Direction
Abstract: In the 2D last passage percolation models (predicted to lie in the KPZ universality class), geodesics are oriented paths moving through random noise, accruing maximum weight. The weights of such geodesics (as their endpoints vary) give rise to a random energy field. It is expected to converge to a universal object known as the Directed Landscape, which is a source of intricate random fractal behaviors. I will talk about recent results on such fractal behaviors, arose from the coupling structure of the maximum weights. The main result is that the difference profile (the difference of the maximum weights from one point to two fixed points, considered as a function on the plane) is non-constant in a set of Hausdorff dimension 5/3. Compared to previous studies, the main challenge is to understand the difference profile in the `temporal direction’. We tackle this by introducing a decomposition of the non-constancy set, constructing the local time process for the geodesic (akin to Brownian local time), and showing that the local time process has Hausdorff dimension 1/3.
This is joint work with Shirshendu Ganguly.
Title: Interface dynamics and conformal maps
Abstract: A range of phenomena in nature can be modelled as a monotone evolution of an interface in the plane, with the dynamics being in some way related to harmonic measure. Examples include (deterministic) Hele-Shaw flow and (random) aggregation models such as DLA. While such models are often very hard to analyze, some progress has been achieved for simplified models using conformal maps and the Loewner differential equation. I will survey some of the models, tools, and results along with open questions. Based in part on joint works with Amanda Turner (Lancaster) and Alan Sola (SU) and with Yilin Wang (MIT).
Title: Around Wilson loops in 4D lattice gauge theories
Abstract: Lattice gauge theories were first considered in the 1970s as regularized (and rigorously defined) lattice approximations of continuum quantum field theories known as Yang-Mills theories, the latter which are still to this day lacking rigorous constructions in most physically relevant settings.
In the last few years there has been a renewed interest in the rigorous analysis of 4D lattice gauge theories in the probability community. I will discuss some background and basic ideas in this area, including Wilson’s pure gauge theory as well as the lattice Higgs model on the Z^4 lattice. I will then report on recent results on the asymptotic behavior of Wilson loop expectations for these models in the simplest settings when the gauge group is finite and abelian. Based on joint work with Malin Forsström (KTH) and Jonatan Lenells (KTH)
Title: Near-critical dimers and massive SLE
Abstract: A program initiated by Makarov and Smirnov is to describe near-critical scaling limits of planar statistical mechanics models in terms of massive SLE and/or Gaussian free field. We consider here the dimer model on the square or hexagonal lattice with doubly periodic weights, which is known to have non Gaussian limits in the whole plane.
In joint work with Levi Haunschmid (TU Vienna) we obtain the following results: (a) we establish a rigorous connection with the massive SLE2 constructed by Makarov and Smirnov; (b) we show that the convergence of the height function in arbitrary bounded domains subject to Temperleyan boundary conditions, and that the scaling limit is universal; and (c) we prove conformal covariance of the scaling limit. Our techniques rely on Temperley's bijection and the "imaginary geometry" approach developed in earlier work with Benoit Laslier and Gourab Ray.
Title: Singularity of discrete random matrices
Abstract: Let Mn be an nxn random matrix whose entries are i.i.d. copies of a discrete random variable ξ. It has been conjectured that the dominant reason for the singularity of Mn is the event that a row or column of Mn is zero, or that two rows or columns of Mn coincide (up to a sign). I will discuss joint work with Ashwin Sah (MIT) and Mehtaab Sawhney (MIT), towards the resolution of this conjecture.
Title: Cutoff for the Asymmetric Riffle Shuffle
Abstract: In the Gilbert-Shannon-Reeds shuffle, a deck of N cards is cut into two approximately equal parts which are riffled
together uniformly at random. This Markov chain famously undergoes total variation cutoff after 3⁄2 log2(N) shuffles. We prove cutoff
for general asymmetric riffle shuffles where the deck is cut into differently sized parts before riffling. The location of the cutoff
was previously conjectured by Lalley.
Title: Fractal Geometry of the KPZ equation
Abstract: The Kardar-Parisi-Zhang (KPZ) equation is a fundamental stochastic PDE related to the KPZ universality class. In this talk,
we focus on how the tall peaks and deep valleys of the KPZ height function grow as time increases. In particular, we will ask what is the appropriate
scaling of the peaks and valleys of the (1+1)-d KPZ equation and whether they converge to any limit under those scaling. These questions will be
answered via the law of iterated logarithms and fractal dimensions of the level sets. The talk will be based on joint works with Sayan Das and Jaeyun Yi.
If time permits, I will also mention an ongoing work with Jaeyun Yi for the (2+1)-d case.
Title: Covering systems of congruences
Abstract: A distinct covering system of congruences is a list of congruences ai ≅ mi, for i = 1, 2, ..., k whose union is the integers. Erdős asked if the least modulus m1 of a distinct covering system of congruences can be arbitrarily large (the minimum modulus problem for covering systems, $1000) and if there exist distinct covering systems of congruences all of whose moduli are odd (the odd problem for covering systems, $25). I'll discuss my proof of a negative answer to the minimum modulus problem, and a quantitative refinement with Pace Nielsen that proves that any distinct covering system of congruences has a modulus divisible by either 2 or 3. The proofs use the probabilistic method. Time permitting, I may briefly discuss a reformulation of our method due to Balister, Bollobás, Morris, Sahasrabudhe and Tiba which solves a conjecture of Shinzel (any distinct covering system of congruences has one modulus that divides another) and gives a negative answer to the square-free version of the odd problem.
Title: Critical percolation on the hierarchical lattice.
Abstract: We consider long-range percolation on the hierarchical lattice, a highly symmetric ultrametric space that serves as a toy model
for the Euclidean lattice Z^d. We will outline how to prove up-to-constants estimates on point-to-point connection probabilities for the
model at criticality and outline several open problems regarding further critical exponents for the model.
Title: Wilson loop expectations as sums over surfaces in 2D
Abstract:
Wilson loop expectations as sums over surfaces in 2D
Although lattice Yang-Mills theory on ℤᵈ is easy to rigorously define, the construction of a satisfactory continuum
theory on ℝᵈ is a major open problem when d ≥ 3. Such a theory should assign a Wilson loop expectation to each suitable
collection ℒ of loops in ℝᵈ. One classical approach is to try to represent this expectation as a sum over surfaces with boundary ℒ.
There are some formal/heuristic ways to make sense of this notion, but they typically yield an ill-defined difference of infinities.
In this talk, I show how to make sense of Yang-Mills integrals as surface sums for d=2, where the continuum theory is already understood.
We also obtain an alternative proof of the Makeenko-Migdal equation and a version of the Gross-Taylor expansion.
Joint work with Joshua Pfeffer, Scott Sheffield, and Pu Yu.
Title: Renormalization group maps for Ising models and tensor networks
Abstract: I review Wilson-Kadanoff type renormalization group (RG) maps for Ising spin systems which
give an approach for studing critical behavior. (These maps are also known as real-space RG transformations.)
These RG maps have been successful numerically, but there are virtually no mathematical results. Even a rigorous
definition of the map seems to be out of reach. The Ising model can be written as a tensor network, and RG maps
can be easily rigorously defined in the tensor network formalism. Numerical studies of such RG maps have been
quite successful at reproducing the known critical behavior in two dimensions. I will review what a tensor network
is and how to define a RG map in this context. Then I will discuss joint work with Slava Rychkov which proves that
in two dimensions for a particular tensor network RG map the high temperature fixed point is locally stable, i.e.,
there is a neighborhood of the high temperature fixed point such that for an initial tensor in this neighborhood,
the iterations of the RG map converge to the high temperature fixed point. We hope that this is a modest first
step towards proving the existence of a non-trivial fixed point for a tensor network RG map which would correspond
to the critical point of the Ising model.
Title: Strong Quantum Unique Ergodicity and its Gaussian fluctuations for Wigner matrices.
Abstract: We prove that the eigenvectors of Wigner matrices satisfy the Eigenstate Thermalisation Hypothesis (ETH),
which is a strong form of Quantum Unique Ergodicity (QUE) with optimal speed of convergence. Then, using this a priori bound
as an input, we analyze the stochastic Eigenstate Equation (SEE) and prove Gaussian fluctuations in the QUE.
The main methods behind the above results are: (i) multi-resolvents local laws established via a novel bootstrap scheme; (ii)
energy estimates for SEE.
Title: p-adic random matrices and particle systems
Abstract: Random p-adic matrices have been studied since the late 1980s as natural models for random groups appearing in number theory and combinatorics.
Recently it has also become clear that the theory has close structural parallels with singular values of complex random matrices, bringing new
techniques from integrable probability and motivating new questions. After outlining this area (no background in p-adic matrices will be assumed),
I will discuss results on the distribution of analogues of singular values for products of many random p-adic matrices. In different regimes we
can prove both Gaussian limits and an intriguing new discrete, Poisson-type local limit—a local interacting particle system on $\mathbb{Z}$ similar
to $q$-TASEP—both appearing quite universally.
The talk will touch on results in https://arxiv.org/abs/2011.09356, https://arxiv.org/abs/2112.03725, and work in progress.
Title: A state space for the 3D Yang-Mills measure
Abstract: In this talk, I will describe some progress towards the construction of the 3D Yang-Mills (YM) measure. In particular, I will introduce a state
space of “distributional gauge orbits” which may possibly support the 3D YM measure. Then, I will describe a result which says that assuming that 3D YM theories
exhibit short distance behavior similar to the 3D Gaussian free field (which is the expected behavior), then the 3D YM measure may be constructed as a probability
measure on the state space. This is based on joint work with Sourav Chatterjee.
Title: TBA
Abstract: TBA
Title: Gibbsian line ensembles and beta-corners processes
Abstract: Gibbs measures are ubiquitous in statistical mechanics and probability theory.
In this talk I will discuss two types of classes of Gibbs measures – random line ensembles and triangular particle arrays,
which have received considerable attention due, in part, to their occurrence in integrable probability.
Gibbsian line ensembles can be thought of as collections of finite or countably infinite independent random walkers
whose distribution is reweighed by the sum of local interactions between the walkers. I will discuss some recent
progress in the asymptotic study of Gibbsian line ensembles, summarizing some joint works with Barraquand, Corwin, Matetski, Wu and others.
Beta-corners processes are Gibbs measures on triangular arrays of interacting particles and can be thought of as
analogues/extensions of multi-level spectral measures of random matrices. I will discuss some recent progress on
establishing the global asymptotic behavior of beta-corners processes, summarizing some joint works with Das and Knizel.
Title: Dimers and embeddings
Abstract:
We introduce a concept of ‘t-embeddings’ of weighted bipartite planar graphs.
We believe that these t-embeddings always exist and that they are good candidates to recover the
complex structure of big bipartite planar graphs carrying a dimer model. We also develop a relevant
theory of discrete holomorphic functions on t-embeddings; this theory unifies Kenyon’s holomorphic
functions on T-graphs and s-holomorphic functions coming from the Ising model.
We provide a meta-theorem on convergence of the height fluctuations to the Gaussian Free Field.
Title: Two types of integrability in Liouville quantum gravity
Abstract: There are two major resources of integrability in Liouville quantum gravity:
conformal field theory and random planar maps decorated with statistical physics models.
I will give a few examples of each type and explain how these two types are compatible.
Recently, cutting and gluing random surfaces in LQG using SLE curves allows us to blend
these two types of integrability to obtain exact results on Liouville conformal field theory,
mating of trees, Schramm-Loewner evolution, and conformal loop ensemble. I will
present a few results in this direction. Based on joint works with Morris Ang, Nina Holden and Guillaume Remy.
Title: Edge Statistics for Lozenge Tilings of Polygons
Abstract: A central feature of random lozenge tilings is that they exhibit boundary induced phase transitions.
Depending on the shape of the domain, they can admit frozen regions and liquid regions. The curve separating these
two phases is called the arctic boundary. For random lozenge tilings of polygonal domains, it is predicted that
the arctic boundary after proper rescaling converges to the Airy process, a universal scaling limit that is
believed to govern various phenomena related to the Kardar–Parisi–Zhang universality class.
In this talk, I will give an overview of this edge universality phenomenon, and explain
a proof for random lozenge tilings of simply connected polygons forbidding specific
(presumably non-generic) behaviors for singular points of the limiting arctic boundary. This is a joint work with Amol Aggarwal.
Title: Integrability of the conformal loop ensemble
Abstract: For 8/3 < κ < 8, the conformal loop ensemble CLEκ is a canonical random ensemble of loops which is
conformally invariant in law, and whose loops locally look like Schramm-Loewner evolution with parameter
κ. It describes the scaling limits of the Ising model, percolation, and other models. When κ ≤ 4 the
loops are simple curves. In this regime we compute the three-point nesting statistic of CLEκ on the
sphere, and show it agrees with the imaginary DOZZ formula of Zamolodchikov (2005). We also obtain
the expression of the (properly normalized) probability that three points are on the same CLE loop in
terms of the DOZZ formula. The analogous quantity for three points on the same cluster was previously
conjectured by Delfino and Viti. To our best knowledge our formula has not been predicted in the physics
literature. Our arguments depend on couplings of CLE with Liouville quantum gravity and the
integrability of Liouville conformal field theory. Based on joint work with Xin Sun.
Title: Stochastic quantization of Yang-Mills
Abstract: We will discuss stochastic quantization of the Yang-Mills model on two and three
dimensional torus. In stochastic quantization we consider the Langevin dynamic for the Yang-Mills
model which is described by a stochastic PDE. We construct local solution to this SPDE and prove that
the solution has a gauge invariant property in law, which then defines a Markov process on the space
of gauge orbits. We will also briefly describe the construction of this orbit space, on which we have
well-defined holonomies and Wilson loop observables. Based on joint work with Ajay Chandra,
Ilya Chevyrev, and Martin Hairer.
Title: The Skew Brownian Permuton
Abstract:
Consider a large random permutation satisfying some constraints or biased according to some statistics.
What does it look like? In this seminar we make sense of this question introducing the notion of permutons.
Permuton convergence has been established for several models of random permutations in various works:
we give an overview of some of these results, mainly focusing on the case of pattern-avoiding permutations.
The main goal of the talk is to present a new family of universal limiting permutons, called skew Brownian
permutons. This family includes (as particular cases) some already studied limiting permutons, such as
the biased Brownian separable permuton and the Baxter permuton. We also show that some natural families
of random constrained permutations converge to some new instances of skew Brownian permutons.
The construction of these new limiting objects will lead us to investigate an intriguing connection
with some perturbed versions of the Tanaka SDE and the SDEs encoding skew Brownian motions.
If time permits, we will present some conjectures on how it should be possible to construct these
new limiting permutons directly from the Liouville quantum gravity decorated with two SLE curves.
Title: Geometric aspects of random matrix ensembles
Abstract: TBA
Title: Empirical measures, geodesic lengths, and a variational formula in first-passage percolation
Abstract: We consider the standard first-passage percolation model on Z^d, in which each edge is assigned an i.i.d. nonnegative weight, and the passage time between any two points is the smallest total weight of a nearest-neighbor path between them. This induces a random ``disordered” geometry on the lattice. Our primary interest is in the empirical measures of edge-weights observed along geodesics in this geometry, say from 0 to [n\xi], where \xi is a fixed unit vector. For various dense families of edge-weight distributions, we prove that these measures converge weakly to a deterministic limit as n tends to infinity. The key tool is a new variational formula for the time constant. In this talk, I will derive this formula and discuss its implications for the convergence of both empirical measures and lengths of geodesics.
Title: Integrability of boundary Liouville CFT
Abstract: Liouville theory is a fundamental example of a conformal field theory (CFT) first introduced in physics by A. Polyakov to describe a canonical random 2d surface. In recent years it has been rigorously studied using probabilistic techniques. In this talk we will study the integrable structure of Liouville CFT on a domain with boundary by proving exact formulas for its correlation functions. Our latest result is derived using conformal welding of random surfaces, in relation with the Schramm-Loewner evolutions. We will also discuss the connection with the conformal blocks of CFT which are fundamental functions determined by conformal invariance that underlie the exact solvability of CFT.
Based on joint works with Morris Ang, Promit Ghosal, Xin Sun, Yi Sun and Tunan Zhu.
Title: TBA
Abstract: TBA
Title: Universality for the min-modulus of random trigonometric polynomials
Abstract: Consider the restriction to the unit circle of a random degree-n polynomial with iid coefficients
(the Kac polynomial). Recent work of Yakir and Zeitouni shows that for Gaussian coefficients, the minimum modulus (suitably rescaled)
follows a limiting exponential distribution. We show this is a universal phenomenon, extending their result to arbitrary sub-Gaussian
coefficients, such as Rademacher signs (Littlewood polynomials). Our approach relates the joint distribution of small values at several
angles to that of a random walk in high-dimensional phase space, for which we obtain strong central limit theorems. The case of discrete
coefficients is particularly challenging as the distribution is sensitive to arithmetic structure among the angles, requiring methods
from additive combinatorics. Based on joint work with Hoi Nguyen.
Title: Tails of the empirical distribution on a geodesic in first-passage percolation
Abstract: First-passage percolation defines a random pseudo-metric on Z^d by attaching to each nearest-neighbor edge of the lattice
a non-negative weight. Geodesics are paths which realize the distance between sites. This project considers the question of what the environment
looks like on a geodesic through the lens of the empirical distribution of the weights on that geodesic. We obtain upper and lower tail bounds
for the upper and lower tails which quantify and limit the intuitive statement that the typical weight on a geodesic should be small compared
to the marginal distribution of an edge weight. Based on joint work-in-progress with Michael Damron, Wai-Kit Lam, and Xiao Shen which was started at the AMS MRC on Spatial Stochastic Models in 2019.
Title: Universality for random band matrices
Abstract: Random band matrices (RBM) are natural intermediate models to study eigenvalue statistics
and quantum propagation in disordered systems, since they interpolate between mean-field type
Wigner matrices and random Schrodinger operators. In particular, RBM can be used to model the Anderson
metal-insulator phase transition (crossover) even in 1d. In this talk we will discuss some recent progress in application of the supersymmetric method (SUSY)
and transfer matrix approach to the analysis of local spectral characteristics of some
specific types of 1d RBM.
Title: A random walk through sub-Riemannian geometry
Abstract: A sub-Riemannian manifold M is a connected smooth manifold such that the only smooth curves in M which are admissible are those whose tangent
vectors at any point are restricted to a particular subset of all possible tangent vectors. Such spaces have applications in control theory and mechanics, as well
as in the study of hypo-elliptic operators. We will construct a random walk on M which converges to a process whose infinitesimal generator is one of the natural
sub-elliptic Laplacian operators. In the elliptic case this construction goes back to M. Pinsky. We will also describe these Laplacians geometrically and discuss
the difficulty of defining one which is canonical in this setting. Examples will be provided. If time permits, we will also talk about discrete time random walks
and large deviations for a class of such degenerate manifolds. This is based on a joint work with Tom Laetsch, Tai Melcher and Jing Wang.
Title: Volume of metric balls in Liouville quantum gravity
Abstract: Liouville quantum gravity (LQG) is a random geometry associated with the planar Gaussian free field.
This geometry was introduced in the physics literature by Polyakov in the 80's and is conjectured to describe the scaling
limit of random planar maps. In this talk, we give an introduction to LQG seen as a metric measure space and discuss results
on the volume of metric balls. Based on a joint work with Morris Ang and Xin Sun.
Title: New results for surface growth
Abstract: The growth of random surfaces has attracted a lot of attention in probability theory in the last ten years, especially in the context of the Kardar-Parisi-Zhang (KPZ)
equation. Most of the available results are for exactly solvable one-dimensional models. In this talk I will present some recent results for models that are not exactly solvable.
In particular, I will talk about the universality of deterministic KPZ growth in arbitrary dimensions, and if time permits, a necessary and sufficient condition for superconcentration
in a class of growing random surfaces that includes variants of ballistic deposition and the restricted solid-on-solid model.
Title: Schrödinger operators with potentials generated by hyperbolic transformations
Abstract: We discuss discrete one-dimensional Schrödinger operators whose potentials are generated by sampling along the orbits of a general hyperbolic transformation and present
results showing that the Lyapunov exponent is positive away from a small exceptional set of energies for suitable choices of the ergodic measure and the sampling function. These results apply
in particular to Schrödinger operators defined over expanding maps on the unit circle, hyperbolic automorphisms of a finite-dimensional torus, and Markov chains. (Joint work with Artur Avila
and Zhenghe Zhang)
Title: Phase Transitions in the Asymptotically Singular Parabolic Anderson Model
Abstract: The Parabolic Anderson Model (PAM) is a model of particle diffusion with branching/killing rates determined by a random environment. In the classical setting, where the random
environment is generated by a “smooth” random function, the behavior of the PAM is well understood. In contrast, when the random environment is generated by a singular noise, much less is known,
as difficult technical problems must be overcome to analyze the PAM in those cases.
In this talk, we discuss a new approach to study the PAM that interpolates between the classical and singular settings. In particular, we show that the some of the features of the PAM with singular
noise can be accessed using only classical tools.
Title: Shattering Phase vs Metastability for Spin Glasses
Abstract: I will report mainly on a recent joint work with Aukosh Jagannath (University of Waterloo) (Arxiv 2104.08299)
Spin glasses are complex models of statistical mechanics, which have been studied and understood in real depth by physicists since the 70's, both in their statics and dynamics properties. I will very
briefly survey the current mathematical understanding in the case of spherical models of spin glasses, using the approach based on the topological complexity of the energy landscapes. This approach has
been shown to be a powerful tool for the low temperature phase of `replica-symmetry breaking', but I will here address questions pertaining to a higher temperature regime in the `replica-symmetric phase'.
In this regime, we find that there are at least two distinct temperatures related to non-trivial behavior. First we prove that there is a regime of temperatures in which the spherical p-spin model exhibits
a shattering phase. We then find that metastable states exist up to an even higher temperature as predicted by Barrat-Burioni-Mezard (1992). This work is based on a Thouless-Anderson-Palmer decomposition
which builds on the work of Eliran Subag(2018). We then present a series of questions and conjectures regarding the sharp phase boundaries for shattering and slow mixing.
Title: Complex Gaussian Multiplicative Chaos
Abstract: In this talk, we investigate the problem of defining a random distribution corresponding
to the exponential of a Gaussian log-correlated field. We consider a Gaussian field indexed by
whose covariance diverges logarithmically on the diagonal ( K(x,y)= -log|x-y|+O(1) ) and γ
a complex number and we try to define a distribution which formally corresponds to the complex exponential of .
This mathematical object is what is called complex Gaussian multiplicative chaos. The field not being a function,
but a random Gaussian distribution, the expression does not make sense literally and a renormalization
procedure is necessary. This problem finds its origin in Kahane's work in the 80s, and found since numerous applications
in theoretical Physics (Conformal Field Theory, Coulomb Gas, Modeling of turbulent flows).
The object changes its nature when γ varies. The complex GMC presents a phase diagram which splits
the complex plane in three regions. We will present results concerning two of these three phases.
Title: Moments of characteristic polynomials of the classical compact groups
Abstract: Moments of characteristic polynomials have connections to log-correlated fields, Toeplitz and Hankel determinants,
combinatorics, and number theory. In this talk, I will introduce `moments of moments' of characteristic polynomials. Our results give
their asymptotic behaviour, answering a conjecture of Fyodorov and Keating. This talk will discuss joint work with Jon Keating and Theo Assiotis.
Title: Markov property and conditional rigidity for 3D Ising interfaces
Abstract: Dobrushin famously showed that the interface of a 3D Ising model with plus boundary conditions in the upper half-space
and minus boundary conditions in the lower half-space is rigid at low temperatures, i.e., its height oscillations are O(1) with exponential
tails. Most of the progress in analyzing simpler models of random surfaces (e.g., solid-on-solid, Gaussian free field) crucially relies on a
Markov property: the law of the surface inside a height-h level curve is independent of the height profile outside the level curve. However,
such a Markov property does not hold for the Ising interface. In joint work with Lubetzky, we established an approximate form of the Markov property,
from which one can e.g., deduce that inside a height h-level curve, the interface is rigid about height-h, and the asymptotics of its recentered
maximum do not depend on the height profile outside the level curve. In this talk we will first recall Dobrushin's proof of rigidity, then describe
the new framework for establishing conditional rigidity estimates, and explore potential applications.
Title: Probabilistic conformal blocks for Liouville CFT on the torus
Abstract: This talk presents a probabilistic construction of 1-point Virasoro conformal blocks on the torus for central charge greater than 25.
These objects appear in the study of Liouville conformal field theory (CFT) and are related to 4-D supersymmetric gauge theory through the AGT correspondence.
I will present our construction using Gaussian multiplicative chaos and give a sketch of the proof, which uses the integrable structure of Liouville CFT.
If time permits, I will mention connections to work in progress on modular symmetry for these conformal blocks. Based on joint work with Promit Ghosal, Guillaume Remy, and Xin Sun.
Title: Large time behaviour of the parabolic Anderson model
Abstract: We consider the parabolic Anderson model with a white noise potential in two dimensions.
This model is also called the stochastic heat equation with a multiplicative noise. We study the large time
asymptotics of the total mass of the solution. Due to the irregularity of the white noise, in two dimensions
the equation is a priori not well-posed. Using paracontrolled calculus or regularity structures one can make
sense of the equation by a renormalisation, which can be thought of as ``subtracting infinity of the potential''.
To obtain the asymptotics of the total mass we use the spectral decomposition, an alternative Feynman-Kac type
representation and heat-kernel estimates which come from joint works with Khalil Chouk, Wolfgang König and Nicolas Perkowski.
Title: Imaginary multiplicative chaos: different questions from different contexts, and a few answers too.
Abstract: Imaginary multiplicative chaos is formally given by exp(iG), where G is a log-correlated Gaussian field in d dimensions.
It comes up in several different contexts. For example
- in its relation to the real multiplicative chaos, that is central in the probabilistic study of Liouville quantum gravity and Liouville CFT,
- when taking the continuum limit of the spin field for the XOR-Ising model,
- in relation to the Kosterlitz-Thouless-type of phase transitions.
In this talk I will try to explain how imaginary chaos comes up in these contexts, which questions it brings along, and how to answer some of these questions. Based on joint works with G. Bavarez, A. Jego, J. Junnila.
Title: Universality in Random Growth Processes
Abstract: Universality in disordered systems has always played a central
role in the direction of research in Probability and Mathematical Physics,
a classical example being the Gaussian universality class (the central
limit theorem). In this talk, I will describe a different universality
class for random growth models, called the KPZ universality class. Since
Kardar, Parisi and Zhang introduced the KPZ equation in their seminal
paper in 1986, the equation has made appearances everywhere from bacterial
growth, fire front, coffee stain to the top edge of a randomized game of
Tetris; and this field has become a subject of intense research interest
in Mathematics and Physics for the last 15 to 20 years. The random growth
processes that are expected to have the same scaling and asymptotic
fluctuations as the KPZ equation and converge to the universal limiting
object called the KPZ fixed point, are said to lie in the KPZ universality
class, though this KPZ universality conjecture has been rigorously proved
for only a handful of models till now. Here, I will talk about some
universal geometric properties of the KPZ fixed point and the underlying
landscape and show that the KPZ equation and exclusion processes converge
to the KPZ fixed point under the 1:2:3 scaling, establishing the KPZ
universality conjecture for these models, which were long-standing open
problems in this field. The talk is based on joint works with Jeremy Quastel, Balint Virag and
Duncan Dauvergne.
Title: Universality for Lozenge Tiling Local Statistics
Abstract: The statistical behavior of random tilings of large domains has been an intense topic of mathematical research for decades,
partly since they highlight a central phenomenon in physics: local behaviors of highly correlated systems can be very sensitive to boundary conditions.
Indeed, a salient feature of random tiling models is that the local densities of tiles can differ considerably in different regions of the domain,
depending on the boundary conditions. Thus, a question of interest, originally mentioned by Kasteleyn in 1961, is how the shape of the domain affects
the local behavior of a random tiling. In this talk, we outline recent work that provides an answer (originally predicted by Cohn-Kenyon-Propp)
to this question for random lozenge tilings of essentially arbitrary domains. The proof will proceed by locally coupling a uniformly random lozenge tiling
with a model of Bernoulli random walks conditioned to never intersect. Central to implementing this procedure is to establish a local law for the random tiling,
which states that the associated height function is approximately linear on any mesoscopic scale.
Title: The spectral edge of (sub-)critical Erdös-Rényi graphs
Abstract: It is well known that the Erdős-Rényi graph on N vertices with edge probability d/N undergoes a dramatic change in behaviour when the mean degree d
crosses the critical scale log(N): the degrees of the graph cease to concentrate about their means and the graph loses its homogeneity. We analyse the eigenvalues and
eigenvectors of its adjacency matrix in the regime where the mean degree d is comparable to or less than the critical scale log(N). We show that the eigenvalue process
near the spectral edges is asymptotically Poisson, and the intensity measure is determined by the fluctuations of the large degrees as well as the size of the 2-spheres
around vertices of large degree. We conclude that in general the laws of the largest eigenvalues are not described by the classical Fisher–Tippett–Gnedenko theorem. As an
application of our result, we prove that the associated eigenvectors are are exponentially localized in unique, disjoint balls. Together with the previously established
complete delocalization of the eigenvectors in the middle of the spectrum, this establishes the coexistence of a delocalized and a localized phase in the critical Erdös-Rényi graph.
Joint work with Johannes Alt and Raphael Ducatez.
Title: Tightness and Brownian regularity for KPZ line ensemble
Abstract: Many important models in integrable probability (e.g. the KPZ equation,
solvable directed polymers, ASEP, stochastic six vertex model) can be embedded
into Gibbsian line ensembles. The Gibbs property provides a powerful resampling
invariance against Brownian bridges over an arbitrary interval. In this talk, I will
explain how to study tightness and path regularity of KPZ line ensemble using
this hidden probabilistic structure.
Title: Mean-field tricritical polymers
Abstract: We consider a random walk on the complete graph. The walk
experiences competing self-repulsion and self-attraction, as well
as a variable length. Variation of the parameters governing
the self-attraction and the variable length leads to a rich phase
diagram containing a tricritical point (known as the "theta" point
in chemical physics). We discuss the phase diagram, as well as
the method of proof used to determine the phase diagram. The method
involves a supersymmetric representation for the random walk,
together with the Laplace method for an integral with large parameter.
This is a joint work with Roland Bauerschmidt, to appear in Probability and
Mathematical Physics.
Title: Branching diffusion processes
Abstract: We investigate the asymptotic behavior of solutions to parabolic partial differential equations (PDEs)
in R^d with space-periodic diffusion matrix, drift, and potential. Using this asymptotics, we describe the behavior of branching
diffusion processes in periodic media. For a super-critical branching process in periodic media, we distinguish two types of behavior
for the normalized number of particles in a bounded domain, depending on the distance of the domain from the region where the bulk
of the particles is located. At distances that grow linearly in time, we observe intermittency (i.e., the k−th moment dominates the
k−th power of the first moment for some k), while, at distances that grow sub-linearly in time, we show that all the moments converge.
Title: Stability and chaos in dynamical last passage percolation
Abstract: Many complex disordered systems in statistical mechanics are characterized by intricate energy landscapes.
The ground state, the configuration with lowest energy, lies at the base of the deepest valley.
In important examples, such as Gaussian polymers and spin glass models, the landscape has many valleys and the abundance of near-ground states
(at the base of valleys) indicates the phenomenon of chaos, under which the ground state alters profoundly when the disorder of the model is slightly perturbed.
In this talk, we will discuss a recent work computing the critical exponent that governs the onset of chaos in a dynamic manifestation of a canonical model in
the Kardar-Parisi-Zhang [KPZ] universality class, Brownian last passage percolation [LPP]. In this model in its static form, semi-discrete polymers advance
through Brownian noise, their energy given by the integral of the white noise encountered along their journey. A ground state is a geodesic, of extremal energy given its endpoints.
We will show that when Brownian LPP is perturbed by evolving the disorder under an Ornstein-Uhlenbeck flow,
for polymers of length n, a sharp phase transition marking the onset of chaos is witnessed at the critical time n^{-1/3}, by showing that the overlap between the geodesics at times
zero and t > 0 that travel a given distance of order n is of order n when t<< n^{-1/3}; and of a smaller order when t>> n^{-1/3}. We expect this exponent to be universal across a wide
range of interface models. The proof relies on Chatterjee's harmonic analytic theory of equivalence of superconcentration and chaos in Gaussian spaces and a refined understanding of the
static landscape geometry of Brownian LPP.
The talk is based on a recent joint work with Alan Hammond (arxiv.org/abs/2010.05837 and the companion paper arxiv.org/abs/2010.05836).
Title: Some generalizations of thick points of Gaussian free fields in any dimension
Abstract: The geometry of log-correlated Gaussian free fields (GFFs) has been extensively studied. In particular, for the 2D GFF in the continuum setting (associated with the Laplace operator on a bounded planar domain
with the Dirichlet boundary condition), Hu, Miller and Peres (2010) studied the thick points, which, heuristically speaking, are locations where the GFF becomes "exceptionally" large, and they determined the Hausdorff dimension of
the set consisting of thick points. We are interested in analyzing finer structures of such exceptional sets. In this talk, we will discuss some extensions of the study of such exceptional behaviors to a more general class of
Gaussian random fields, for which we will continue using the terminology "GFFs". We adopt a sphere averaging regularization to study polynomial-correlated GFFs in R^d for d>=3, and conduct an analysis of the "thick point" analogous
to the one in the log-correlated setting. We propose a general framework to study certain types of exceptional behaviors for both log-correlated and polynomial-correlated GFFs. In addition to reproducing the existing results on thick
point sets, this framework also gives rise to new exceptional sets, the study of which offers new information on the random geometry of GFFs.
Title: Liouville quantum gravity with matter central in (1,25): a probabilistic approach
Abstract: Liouville quantum gravity (LQG) is a theory of random fractal surfaces with origin in the physics literature in the 1980s. Most literature is about
LQG with matter central charge c∈(-∞,1]. We study a discretization of LQG which makes sense for all c∈(-∞,25). Based on a joint work with Gwynne, Pfeffer, and Remy.
Title: Bigeodesics and the density of the geodesic tree in first-passage percolation
Abstract: In first-passage percolation, one assigns random nonnegative weights to the edges of Z^d and considers the resulting weighted graph metric. Many authors have studied the question of
existence of "bigeodesics": doubly infinite geodesics for this metric, with most work in the case d = 2 (where bigeodesics have been ruled out in certain exactly solvable models). We will present the first
progress on this question for d >= 3 under no unproven assumptions. We will also discuss the resolution of the "highways and byways" conjecture of Hammersley-Welsh, showing in a sense that the density of
points lying in geodesics containing the origin is zero.
Title: SLE, energy duality, and foliations by Weil-Petersson quasicircles
Abstract: The Loewner energy for Jordan curves first arises from the small-parameter large deviations of Schramm-Loewner evolution (SLE). It is finite if and only if the curve is a Weil-Petersson
quasicircle, an interesting class of Jordan curves appearing in Teichmuller theory, geometric function theory, and string theory with currently more than 20 equivalent definitions. In this talk, I will show
that the large-parameter large deviations of SLE gives rise to a new Loewner-Kufarev energy, which is dual to the Loewner energy via foliations by Weil-Petersson quasicircles and exhibits remarkable features
and symmetries. Based on joint works with Morris Ang and Minjae Park (MIT) and with Fredrik Viklund (KTH).
Title: Spectral Statistics of Lévy Matrices
Abstract: Lévy matrices are symmetric random matrices whose entries are independent \alpha-stable laws. Such distributions have infinite variance, and when \alpha <1, infinite mean. In the latter case these matrices are conjectured to exhibit a sharp transition from a delocalized regime at low energy to a localized regime at high energy, like the infamous Anderson model in mathematical physics. We discuss work establishing the existence of a delocalized regime with GOE eigenvalue statistics. Further, we characterize the eigenvector statistics in this regime and find they display novel, non-Gaussian behavior.
These describe joint works with Amol Aggarwal, Jake Marcinek, and Horng-Tzer Yau.
Probability seminar:
Joshua Pfeffer - MIT
2:30-3:30 PM, Eckhart 202
Title: Understanding Liouville quantum gravity through two square subdivision models
Abstract: In my talk I will discuss a general approach to better understand the geometry of Liouville quantum gravity (LQG). The idea, roughly speaking, is to partition the random surface into dyadic squares of roughly the same "LQG size''. Based on this approach, I will introduce two different models of LQG that will provide answers to three questions in the field:
1. Rigorously explain the so-called "DDK ansatz'' by proving that, for a surface with metric tensor some regularized version of the LQG heuristic metric tensor, its law corresponds to sampling a surface with probability proportional to the $(-c/2)$-th power of the zeta-regularized determinant of the Laplacian, with $c$ the matter central charge.
2.
Provide a heuristic picture of the geometry of LQG with matter central charge in the interval $(1,25)$. (The geometry in this regime is mysterious even from a physics perspective.)
3. Explain why many works in the physics literature may have missed the nontrivial conformal geometry of LQG with matter central charge in the interval $(1,25)$ when they suggest (based on numerical simulations and heuristics) that LQG exhibits the macroscopic behavior of a continuum random tree in this phase.
This talk is based on a joint work with Morris Ang, Minjae Park, and Scott Sheffield; and a joint work with Ewain Gwynne, Nina Holden, and Guillaume Remy.
Statistics Colloquium: Yi Sun - Columbia
4:00-5:00 PM, Jones 303
Title: Fluctuations for Products of Random Matrices
Abstract: Products of large random matrices appear in many modern applications such ashigh dimensional statistics (MANOVA estimators), machine learning (Jacobians of neural networks), and population ecology (transition matrices of dynamical systems). Inspired by these situations, this talk concerns global limits and fluctuations of singular values of products of independent random matrices as both the size N and number M of matrices grow. As N grows, I will show for a variety of ensembles that fluctuations of the Lyapunov exponents converge to explicit Gaussian fields which transition from log-correlated for fixed M to having a white noise component for M growing with N. I will sketch our method, which uses multivariate generalizations of the Laplace transform based on the multivariate Bessel function from representation theory.
Title: Random walks in random environment on the strip
Abstract: The Random walks in random environment on the strip model was introduced by
Bolthausen and Goldsheid in order to treat 1 dimensional Random walks
in random
environment with bounded jump. In this talk I will review recent
results about this model.
Title: Community Detection in Sparse Random Hypergraphs
Abstract: The stochastic block model is a generative model for random graphs with a community structure, which has been one of the most fruitful research topics in community detection and clustering. A phase transition behavior for detection was conjecured by Decelle et al. (2011) at the Kesten-Stigum threshold, and was confirmed by Mossel et al. (2012, 2013) and Massoulié (2013). We consider the community detection problem in random hypergraphs. Angelini et al. (2015) conjectured a detection threshold in sparse hypergraphs generated by a hypergraph stochastic block model (HSBM). We confirmed the positive part of the phase transition by a generalization of the method developed in Massoulié (2013). We introduced a matrix which counts self-avoiding walks on hypergraphs, whose leading eigenvectors give us a correlated reconstruction of the community. In the course of proving our main result, we developed a moment method for sparse hypergraphs and constructed a coupling between the local neighborhood of HSBMs and multitype Poisson Galton-Watson hypertrees. This is joint work with Soumik Pal.
Title: Title: A combinatorial approach to the quantitative invertibility of random matrices.
Abstract: Let $s_n(M_n)$ denote the smallest singular value of an $n\times n$ random matrix $M_n$. We will discuss a novel combinatorial approach (in particular, not using either inverse Littlewood--Offord theory or net arguments) for obtaining upper bounds on the probability that $s_n(M_n)$ is smaller than $\eta \geq 0$ for quite general random matrix models. Such estimates are a fundamental part of the non-asymptotic theory of random matrices and have applications to the strong circular law, numerical linear algebra etc. In several cases of interest, our approach provides stronger bounds than those obtained by Tao and Vu using inverse Littlewood--Offord theory.
Title: Roots of random functions
Abstract: The study of random functions has been investigated for many decades. In this talk, we will discuss several classical results by Kac, Littlewood-Offord, Erdos-Offord, etc. together with recent developments and open problems in the field. We also discuss a general framework to prove universality results for correlation functions of the roots and apply it to study various questions on random polynomials, random trigonometric functions, and random eigenfunctions. Using these universality results, we estimate the number of nodal intersections and the number of real roots. We also show that the number of real roots satisfies the Central Limit Theorem.
This talk is based on several joint papers with Mei-Chu Chang, Yen Do, Hoi Nguyen, and Van Vu.
Title: Asymptotics of discrete β-corners processes via discrete loop equations
Abstract: We introduce and study stochastic particle ensembles which are natural discretizations of general β-corners processes. We prove that under technical assumptions on a general analytic potential the global fluctuations for the difference between two adjacent levels are asymptotically Gaussian. The covariance is universal and remarkably differs from its counterpart in random matrix theory. Our main tools are certain novel algebraic identities that are multi-level analogues of the discrete loop equations. Based on joint work with Evgeni Dimitrov (Columbia University)
Title: Large deviations of subgraph counts for sparse random graphs.
Abstract: In this talk, based on joint works with Nick Cook and with Sohom Bhattacharya, I will
discuss recent developments in the emerging theory of nonlinear large deviations,
focusing on sharp upper tails for counts of several fixed subgraphs in a large sparse
random graph, such as Erdos–Renyi or uniformly d-regular. Time permitting, I will
describe our quantitative versions of the regularity and counting lemmas, which are
geared for the study of sparse random graphs in the large deviations regime,
and what our results suggest regarding certain questions in extremal graph theory.
Title: Distorted Brownian motion on spaces with varying dimension
Abstract: We introduce "distorted Brownian motion" (dBM) on a state space with varying dimension. Roughly speaking, the state space consists of two components: a 3-dimensional component and a 1-dimensional component. These two parts are joined together at the origin. The restriction of dBM on the 3-d component models a homopolymer with attractive potential at the origin. The restriction of dBM on the 1-d component also receives a push towards the origin. Such a process can be nicely characterized in terms of Dirichlet form, and we can find its density estimate by characterizing its radial process.
Title: Fermionic eigenvector moment flow
Abstract: We first present known results and open problems on the study of eigenvector statistics of large random matrices such as complete delocalization, quantum unique ergodicity or asymptotic entry distribution. We willl see how we can obtain some of these properties dynamically using the Dyson Brownian motion and the Bourgade-Yau eigenvector moment flow: it consists of a parabolic equation followed by eigenvector moments. We will then present new moment observables which follow a similar equation and which can be seen as a Fermionic counterpart to the (Bosonic) original ones. By combining the information obtained through the study of these two families of observables, we can compute, previously intractable, correlations between eigenvectors.
Title: Existence and uniqueness of the Liouville quantum gravity metric for $\gamma \in (0,2)
Abstract: We show that for each $\gamma \in (0,2)$, there is a unique metric associated with $\gamma$-Liouville quantum gravity (LQG). More precisely, we show that for the Gaussian free field $h$ on a planar domain $U$, there is a unique random metric $D_h = ``e^{\gamma h} (dx^2 + dy^2)"$ on $U$ which is uniquely characterized by a list of natural axioms.
The $\gamma$-LQG metric can be constructed explicitly as the scaling limit of \emph{Liouville first passage percolation} (LFPP), the random metric obtained by exponentiating a mollified version of the Gaussian free field. Earlier work by Ding, Dub\'edat, Dunlap, and Falconet (2019) showed that LFPP admits non-trivial subsequential limits. We show that the subsequential limit is unique and satisfies our list of axioms. In the case when $\gamma = \sqrt{8/3}$, our metric coincides with the $\sqrt{8/3}$-LQG metric constructed in previous work by Miller and Sheffield.
We also present several open problems related to the LQG metric.
Based on four joint papers with Jason Miller, one joint paper with Julien Dubedat, Hugo Falconet, Josh Pfeffer, and Xin Sun, and one joint paper with Josh Pfeffer.
Title: Multiple SLEs, discrete interfaces, and crossing probabilities
Abstract: Multiple SLEs are conformally invariant measures on families of curves, that naturally correspond to scaling limits of interfaces in critical planar lattice models with alternating (â€Âgeneralized Dobrushinâ€Â) boundary conditions. I discuss classification of these measures and how the convergence for discrete interfaces in many models is obtained as a consequence. When viewed as measures with total mass, the multiple SLEs can also be related to probabilities of crossing events in lattice models.
The talk is based on joint works with
Hao Wu (Yau Mathematical Sciences Center, Tsinghua University)
and Vincent Beffara (Université Grenoble Alpes, Institut Fourier).
Title: On the range of lattice models in high dimensions.
Abstract: We investigate the scaling limit of the {\em range} (the set
of visited vertices) for a general class of critical lattice models,
starting from a single initial particle at the origin. Conditions
are given on the random sets and an associated ``ancestral relation''
under which, conditional on long term survival, the rescaled ranges
converge weakly to the range of super-Brownian motion as random
sets. These hypotheses also give precise asymptotics for the limiting
behaviour of the probability of exiting a large ball. Applications
include voter models, contact processes, oriented percolation and lattice
trees. This is joint work with Mark Holmes and also features work
of Akira Sakai and Gordon Slade.
Title: A Tale of Two Integrals
Abstract: I will discuss the asymptotic behavior of two matrix integrals: the Harish-Chandra/Itzykson-Zuber integral, and its additive counterpart, the Brezin-Gross-Witten integral. Both are integrals over the group U(N), and their behavior in the large N limit is the subject of a pair of conjectures formulated by physicists in 1980 in connection with the large N limit in lattice gauge theory. I will discuss a proof of these conjectures which involves relating them to fundamental combinatorial structures, in particular Hurwitz numbers and increasing subsequences in permutations.
Title: Scaling limit of uniform spanning tree in three dimensions
Abstract: In the talk, we will show the existence of the scaling limit of three-dimensional uniform spanning tree (UST) with respect to the Gromov-Hausdorff-Prohorov type topology and obtain several properties of the limiting tree. Moreover, we will prove that the rescaled simple random walk on the 3D UST converges weakly to a diffusion on the limit tree above. Detailed transition density estimates for the limiting process will be derived. These are ongoing works with O. Angel (UBC), D. Croydon (Kyoto University) and S. Hernandez Torres (UBC).
Title: Homogenization of a class of one-dimensional nonconvex viscous Hamilton-Jacobi equations with random potential
Abstract: I will present joint work with Elena Kosygina and Ofer Zeitouni in which we prove the homogenization of a class of one-dimensional viscous Hamilton-Jacobi equations with random Hamiltonians that are nonconvex in the gradient variable. Due to the special form of the Hamiltonians, the solutions of these PDEs with linear initial conditions have representations involving exponential expectations of controlled Brownian motion in a random potential. The effective Hamiltonian is the asymptotic rate of growth of these exponential expectations as time goes to infinity and is explicit in terms of the tilted free energy of (uncontrolled) Brownian motion in a random potential. The proof involves large deviations, construction of correctors which lead to exponential martingales, and identification of asymptotically optimal policies.
Title: Laplacian growth and sandpiles on the Sierpinski gasket: limit shape universality, fluctuations, and beyond
Abstract: Given a locally finite connected graph with a distinguished vertex $o$, start $m$ particles at $o$, and let them aggregate according to one of the four following discrete Laplacian growth models: internal diffusion-limited aggregation (IDLA), rotor-router aggregation (RRA), divisible sandpiles, and abelian sandpiles. We are interested in describing the limit shapes and radial growth of the cluster in each model. Do they coincide for all 4 models? And how sharp a radial bound can one get?
On the Euclidean lattice: No, though there are many results on radial bounds. On the Sierpinski gasket graph, where $o$ is the corner vertex: Yes, and sharp radial bounds are now available.
My talk will address the gasket story and consists of two parts:
1) By solving the divisible sandpile problem, one can gain access to information about the harmonic measure and obtain limit shape results for IDLA and RRA. I will describe how the proofs work on the gasket, incorporating ideas from the earliest IDLA proofs (Lawler-Bramson-Griffeath), random walks on graphs (Harnack inequality), and a fast simulation algorithm (Friedrich-Levine).
2) The abelian sandpile problem poses an entirely different set of challenges. Quite luckily, on the gasket we have solved the abelian sandpile growth problem EXACTLY. The key idea is to exploit the cellular structure, the cut point structure, and axial symmetries to perform systematic topplings in waves. This allows us to inductively establish tiling patterns in the sandpile configurations, in particular, the identity elements of the associated sandpile groups. It also leads to the enumeration of all the radial jumps in the growing cluster, and implies, via the renewal theorem, a radial asymptotic formula in the form of a power law modulated by log-periodic oscillations---the best possible result on such a state space.
Based on joint works with Wilfried Huss, Ecaterina Sava-Huss (TU Graz), Alexander Teplyaev (UConn), and Jonah Kudler-Flam (UChicago).
Title: Dynamical percolation on random triangular lattices
Abstract:
Dynamical (site) percolation on a graph is a Markov process where the state space is the set of possible black/white colorings of the vertices of the graph. Each vertex is associated with an independent Poisson clock, and the color of a vertex is resampled every time its clock rings. Dynamical percolation on the regular triangular lattice was thoroughly studied by Garban, Pete and Schramm. In this talk we will discuss the case when the graph is a uniformly sampled triangulation. In particular, we will explain how to describe the scaling limit of this process and show its ergodicity. Time permitting, we will also explain the role it played in the study of the conformal structure of uniform triangulations. Based on joint work with Christophe Garban, Nina Holden and Avelio Sepulveda.
Title: Localization phenomena of directed polymers
Abstract: On the d-dimensional integer lattice, directed polymers are paths of a random walk that have been reweighted according to a random environment that refreshes at each time step. The qualitative behavior of the system is governed by a temperature parameter; if this parameter is small, the environment has little effect, meaning all possible paths are close to equally likely. If the parameter is made large, however, the system undergoes a phase transition beyond which the path ''localizes,'' meaning the polymer measure concentrates. In this talk, I will discuss different quantitative statements of this phenomenon, methods of proving these statements, and natural connections to other statistical mechanical systems. (Joint work with Sourav Chatterjee)
Title: The effect of random modifications on Loewner hulls
Abstract: Loewner hulls are determined by their real-valued driving functions, via the Loewner differential equation. We will discuss two projects which study the geometric effect on the Loewner hulls of random modifications. In the first project, which is joint work with Kei Kobayashi and Andrew Starnes, the driving function is composed with a random time change, such as the inverse of an $\alpha$-stable subordinator. In contrast to Schramm-Loewner evolution (SLE), we show that for a large class of random time changes, the time-changed Brownian motion process does not generate a simple curve. Further we develop criteria which can be applied in many situations to determine whether the Loewner hull generated by a time-changed driving function is simple or non-simple. In the second project, joint with Nathan Albin and Pietro Poggi-Corradini, we look at the scaling limit of Peano curves associated to so-called fair spanning trees. In contrast to the convergence of the uniform spanning tree Peano curve to SLE(8), we show that the scaling limit is a deterministic object.
Title: A geometric property for optimal paths and its applications in first passage percolation
Abstract: We consider the first passage percolation model in Z^d with a weight distribution F for 0 < F(0) < p_c. In this paper, we derive a geometric property for optimal paths to show that all of them have to pass an M-exit. By this property, we show that the shape is strictly convex, and we solve the height problem.
Title: Unique continuation and localization on the planar lattice
Abstract: I will discuss joint work with Jian Ding in which we establish localization near the edge for the Anderson Bernoulli model on the two dimensional lattice. Our proof follows the program of Bourgain-Kenig and uses a new unique continuation result inspired by Buhovsky-Logunov-Malinnikova-Sodin.
Title: Universality and delocalization of band random matrix
Abstract: In this talk, we discuss some recent work related to a main conjecture on random matrix theory, i.e. phase-transition conjecture on random matrix theory.
The prediction says that phase-transition occurs at the band width regime W \sim N^{1/2}. For high dimensional matrix, i.e. x,y\in Z^d, H_{xy}, there exists some similar stimulation results.
Based on the development of studying on resolvent, i.e., G=(H-z)^{-1}, we obtained some results on low and high dimension cases. In this talk, we will introduce these work and the main ideas and tools used in these work.
They are jointed work with Laci Erdos, Paul Bourgade, H.T. Yau, Yang Fan, etc.
Title: A support theorem for SLE curves
Abstract: SLE curves are an important family of random curves in the plane. They share many similarities with solutions of SDE (in particular, with Brownian motion). Any question asked for the latter can be asked for the former. Inspired by that, Yizheng and I investigate the support for SLE curves. In this talk, I will explain our theorem with more motivation and idea.
Title: Large concentration of harmonic measure: discrete and continuous case.
Abstract: I will discuss the sharp bounds on the multifractal spectrum of planar harmonic measure and their continuous analogues. In particular, I will talk about the refinements of Beurling estimates on the concentration of harmonic measure.
Title: Invertibility of adjacency matrices for random d-regular graphs
Abstract: The singularity problem of random matrices asks the probability that a given discrete random matrix is singular. The first such result was obtained by Komlós in 1967. He showed a Bernoulli random matrix is singular with probability o(1). This question can be reformulated for the adjacency matrices of random graphs, either directed or undirected. The most challenging case is when the random graph is sparse. In this talk, I will prove that for random directed and undirected d-regular graphs, their adjacency matrices are invertible with high probability for all d>=3. The idea is to study the adjacency matrices over a finite field, and the proof combines a local central limit theorem and a large deviation estimate.
Title: The directed landscape
Abstract: The longest increasing subsequence in a random permutation, the second
class particle in TASEP, and semi-discrete polymers at zero temperature
have the same scaling limit: a random function with Holder exponent
2/3-. This limit can be described in terms of the directed landscape, a
random metric at the heart of the Kardar-Parisi-Zhang universality class.
Joint work with Duncan Dauvergne and Janosch Ortmann.
Title: Chemical distance for level-set percolation of planar metric-graph Gaussian free field
Abstract: We consider percolation of the level-sets of a metric-graph Gaussian free field on a box of the planar integer lattice. We provide an upper bound for the length of the shortest path joining the boundary components of a macroscopic annulus. The bound holds with high probability conditioned on connectivity and is sharp up to a poly-logarithmic factor with exponent one-quarter. This is joint work with Jian Ding.
Title: Bismut's Interpolation Between Riemannian Brownian Motion and Geodesic Flow
Abstract: A few years ago Bismut found a natural family of diffusion processes that interpolates between Brownian motion and the geodesic flow
on a compact Riemannian manifold. The convergence to Brownian motion (together with an attendant Gaussian field) at one end
of the parameter interval is nontrivial and proved in the weak sense of finite dimensional marginal distributions. By slightly extending the traditional
setting for stochastic calculus on manifold we show that the convergence can be realized as a strong one in the path space. Such a more precise
convergence to Brownian motion will help us in understanding asymptotic behavior of some classical functional inequalities for path spaces.
Title: Random walks with local memory
Abstract: The theme of this talk is walks in a random environment of ``signposts'' altered by the walker. I'll focus on three related examples:
1. Rotor walk on Z^2. Your initial signposts are independent with the uniform distribution on {North,East,South,West}. At each step you rotate the signpost at your current location clockwise 90 degrees and then follow it to a nearest neighbor. Priezzhev et al. conjectured that in n such steps you will visit order n^{2/3} distinct sites. I'll outline an elementary proof of a lower bound of this order. The upper bound, which is still open, is related to a famous question about the path of a light ray in a grid of randomly oriented mirrors. This part is joint work with Laura Florescu and Yuval Peres.
2. p-rotor walk on Z. In this walk you flip the signpost at your current location with probability 1-p and then follow it. I'll explain why your scaling limit will be a Brownian motion perturbed at its extrema. This part is joint work with Wilfried Huss and Ecaterina Sava-Huss.
3. p-rotor walk on Z^2. Rotate the signpost at your current location clockwise with probability p and counterclockwise with probability 1-p, and then follow it. This walk ``organizes'' its environment by destroying cycles of signposts. A native environment -- stationary in time, from your perspective as the walker -- is an orientation of the uniform spanning forest, plus one additional edge. This part is joint work with Swee Hong Chan, Lila Greco, and Peter Li.
Title: 2-RSB for spherical mixed p-spin models at zero temperature
Abstract: Historically, the method of replica symmetry breaking (RSB) was introduced by Parisi to study mean field spin glass models. This method has played indispensable role in the physical deduction of Parisi formula. Mathematically, the level of RSB corresponds
to the number of points in the support of the Parisi measure. There have been known Replica Symmetric, 1-step RSB and Full-step RSB 'natural' spin glass models. However, k-step RSB for finite k>1 is less well understood. In this talk, I will show that certain
spherical mixed p-spin glass models are 2-step RSB at zero temperature and present consequences in the energy landscape.
This talk is based on joint work with Antonio Auffinger.
Title: Natural parametrization for the scaling limit of loop-erased random walk in three dimensions
Abstract: We will consider loop-erased random walk (LERW) and its scaling limit in three dimensions. Gady Kozma (2007) shows that as the lattice spacing becomes finer, LERW in three dimensions converges weakly to a random compact set with respect to the Hausdorff distance. We will show that 3D LERW parametrized by renormalized length converges in the lattice size scaling limit to Kozma's scaling limit parametrized by some suitable measure on it with respect to the uniform norm. This is based on joint works with Xinyi Li (University of Chicago).
Title: Percolation on random trees (joint work with Marcus Michelen and Josh Rosenberg)
Abstract: Let T be a tree chosen from Galton-Watson measure and
let {U_v} be IID uniform [0,1] random variables associated
with the edges between each vertex and its parent.
These define coupled Bernoulli percolation processes,
as well as an invasion percolation process.
We study quenched properties of these percolations:
properties conditional on T that hold for almost every T.
The invasion cluster has a backbone decomposition
which is Markovian if you put on the right blinders.
Under suitable moment conditions, the law of the
(a.s. unique) backbone ray is absolutely continuous
with respect to limit uniform measure.
The quenched survival probabilities are smooth in the
supercritical region p > p_c. Their behavior as p -> p_c
depends on moment assumptions for the offspring distribution.
Title: Quantitative CLTs for random walks in random environments
Abstract:
The Berry-Esseen estimates give quantitative error estimates on the CLT for sums of i.i.d. random variables, and the polynomial decay rate for the error depends on moment bounds of the i.i.d. random variables with the optimal $1/\sqrt{n}$ rate of convergence obtained under a third moment assumption. In this talk we will prove quantitative error bounds for CLTs of random walks in random environments (RWRE). That is, for certain models of RWRE it is known that the position of the random walk has a Gaussian limiting distribution, and we obtain quantitative error estimates on the rate of convergence to the Gaussian distribution for such RWRE. This talk is based on joint works with Sungwon Ahn and Xiaoqin Guo.
Title: Algorithmic Pirogov-Sinai theory
Abstract: We develop efficient algorithms to approximate the partition function and sample from the hard-core and Potts models on lattices at sufficiently low temperatures in the phase coexistence regime. In contrast, the Glauber dynamics are known to take exponential time to mix in this regime. Our algorithms are based on the cluster expansion and Pirogov-Sinai theory, classical tools from statistical physics for understanding phase transitions, as well as Barvinok's approach to polynomial approximation. Joint work with Tyler Helmuth and Guus Regts.
Title: Yaglom-type limit theorems for branching Brownian motion with absorption
Abstract: We consider one-dimensional branching Brownian motion in which particles are absorbed at the origin. We assume that when a particle branches, the offspring distribution is supercritical, but the particles are given a critical drift towards the origin so that the process eventually goes extinct with probability one. We establish precise asymptotics for the probability that the process survives for a large time t, improving upon a result of Kesten (1978) and Berestycki, Berestycki, and Schweinsberg (2014). We also prove a Yaglom-type limit theorem for the behavior of the process conditioned to survive for an unusually long time, which also improves upon results of Kesten (1978). An important tool in the proofs of these results is the convergence of branching Brownian motion with absorption to a continuous state branching process.
Title: Phase transitions of random constraint satisfaction problems
Abstract: Random constraint satisfaction problems encode many interesting questions in the study of random graphs such as the chromatic and independence numbers. Ideas from statistical physics provide a detailed description of phase transitions and properties of these models. We will discuss the one step replica symmetry breaking transition that many such models undergo.
Title: Coalescence of polymers in Last Passage Percolation
Abstract:
We will discuss new bounds on the time for polymers in first passage percolation to coalesce. As a consequence we will prove that there are no non-trivial bi-geodesics. The methods will combine bounds given by exactly solvable calculations with tools from percolation theory.
Title: Stability of the elliptic Harnack Inequality
Abstract: Following the work of Moser, as well as de Giorgi and Nash,
Harnack inequalities have proved to be a powerful tool in PDE as well as in
probability. In the early 1990s Grigor'yan and Saloff-Coste
gave a characterisation of the parabolic Harnack inequality (PHI).
This characterisation implies that the PHI is stable under bounded perturbation
of weights, as well as rough isometries. In this talk we prove
the stability of the EHI. The proof uses the concept of a quasi symmetric
transformation of a metric space, and the introduction of these ideas to
Markov processes suggests a number of new problems.
This is joint work with Mathav Murugan (UBC).
Title: Factorizations and estimates of Dirichlet heat kernels for non-local operators with critical killings
Abstract: In this talk I will discuss heat kernel estimates for critical perturbations of non-local operators. To be more precise, let $X$ be the reflected $\alpha$-stable process in the closure of a smooth open set $D$, and $X^D$ the process killed upon exiting $D$. We consider potentials of the form $\kappa(x)=C\delta_D(x)^{-\alpha}$ with positive $C$ and the corresponding Feynman-Kac semigroups. Such potentials do not belong to the Kato class. We obtain sharp two-sided estimates for the heat kernel of the perturbed semigroups. The interior estimates of the heat kernels have the usual $\alpha$-stable form, while the boundary decay is of the form $\delta_D(x)^p$ with non-negative $p\in [\alpha-1, \alpha)$ depending on the precise value of the constant $C$. Our result recovers the heat kernel estimates of both the censored and the killed stable process in $D$. Analogous estimates are obtained for the heat kernel of the Feynman-Kac semigroup of the $\alpha$-stable process in ${\mathbf R}^d\setminus \{0\}$ through the potential $C|x|^{-\alpha}$. All estimates are derived from a more general result described as follows: Let $X$ be a Hunt process on a locally compact separable metric space in a strong duality with $\widehat{X}$. Assume that transition densities of $X$ and $\widehat{X}$ are comparable to the function $\widetilde{q}(t,x,y)$ defined in terms of the volume of balls and a certain scaling function. For an open set $D$ consider the killed process $X^D$, and a critical smooth measure on $D$ with the corresponding positive additive functional $(A_t)$. We show that the heat kernel of the the Feynman-Kac semigroup of $X^D$ through the multiplicative functional $\exp(-A_t)$ admits the factorization of the form ${\mathbf P}_x(\zeta >t)\widehat{\mathbf P}_y(\widehat{\zeta}>t)\widetilde{q}(t,x,y)$. This is joint work with Soobin Cho, Panki Kim and Zoran Vondracek.
Title: Ergodic theory of the stochastic Burgers equation
Abstract: The stochastic Burgers equation is one of the basic evolutionary SPDEs related to fluid dynamics and KPZ, among other things. The ergodic properties of the system in the compact space case were understood in 2000's. With my coauthors, Eric Cator, Kostya Khanin, Liying Li, I have been studying the noncompact case. The one force - one solution principle has been proved for positive and zero viscosity. The analysis is based on long-term properties of action minimizers and polymer measures. The latest addition to the program is the convergence of infinite volume polymer measures to Lagrangian one-sided minimizers in the limit of vanishing viscosity (or, temperature) which results in the convergence of the associated global solutions and invariant measures.
Title: Local density estimate for a hypoelliptic SDE
Abstract: In a series of three papers in the 80’s, Kusuoka and Stroock developed a probabilistic program in order to obtain sharp bounds for the density function of a hypoelliptic SDE driven by a Brownian motion. We aim to investigate how their method can be used to study rough SDEs driven by fractional Brownian motions. In this talk, I will outline Kusuoka and Stroock’s approach and point out where the difficulties are in our current setting.
The talk is based on an ongoing project with Xi Geng and Samy Tindel.
Title: The weight, geometry and coalescence of scaled polymers in Brownian last passage percolation
Abstract: In last passage percolation (LPP) models, a random environment in the two-dimensional integer lattice consisting of independent and identically distributed weights is considered. The weight of an upright path is said to be the sum of the weights encountered along the path. A principal object of study are the polymers, which are the upright paths whose weight is maximal given the two endpoints. Polymers move in straight lines over long distances with a two-thirds exponent dictating fluctuation. It is natural to seek to study collective polymer behaviour in scaled coordinates that take account of this linear behaviour and the two-third exponent-determined fluctuation.
We study Brownian LPP, a model whose integrable properties find an attractive probabilistic expression. Building on a study arXiv:1609.02971
concerning the decay in probability for the existence of several near polymers with common endpoints, we demonstrate that the probability that there exist k disjoint polymers across a unit box in scaled coordinates has a superpolynomial decay rate in k.
This result has implications for the Brownian regularity of the scaled polymer weight profile begun from rather general initial data.
Title: Stationary Harmonic Measure and DLA in the Upper Half Plane
Abstract: In this talk, we introduce the stationary harmonic measure in the upper half plane. By bounding this measure, we are able to define both the discrete and continuous time diffusion limit aggregation (DLA) in the upper half plane with absorbing boundary conditions. We prove that for the continuous model the growth rate is bounded from above by $o(2+\epsilon)$. When time is discrete, we also prove a better upper bound of $o(2/3+\epsilon)$, on the maximum height of the aggregate at time $n$.
Title: Geodesics in First-Passage Percolation
Abstract: First-passage percolation is a classical random growth model which
comes from statistical physics. We will discuss recent results about
the relationship between the limiting shape in first passage
percolation and the structure of the infinite geodesics. This includes
a solution to the midpoint problem of Benjamini, Kalai and Schramm.
This is joint work with Daniel Ahlberg.
Title: Shifted weights and restricted path length in first-passage percolation
Abstract: First-passage percolation has remained a challenging field of study since its introduction in 1965 by Hammersley and Welsh. There are many outstanding open problems. Among these are properties of the limit shape and the Euclidean length of geodesics. This talk describes a convex duality between a shift of the edge weights and the length of the geodesic, together with related results on the regularity of the limit shape as a function of the shift. The talk is based on joint work with Arjun Krishnan (Rochester) and Firas Rassoul-Agha (Utah).
Title: The giant component in a degree-bounded process
Abstract: Graph processes $(G(i),i\ge 0)$ are usually defined as follows. Starting from the empty graph on $n$ vertices, at each step $i$ a random edge is added from a set of available edges. For the $d$-process, edges are chosen uniformly at random among all edges joining vertices of current degree at most $d-1$.
The fact that, during the process, vertices become 'inactive' when reaching degree $d$ makes the process depend heavily on its history. However, it shares several qualitative properties with the classical Erdos-Renyi graph process. For example, there exists a critical time $t_c$ at which a giant component emerges, whp (that is, the largest component in $G(tn)$ goes from logarithmic to linear order).
In this talk we consider $d\ge 3$ fixed and describe the growth of the size of the giant component. In particular, we show that whp the largest component in $G((t_c+\eps)n)$ has asymptotic size $cn$, where $c\sim c_d \eps$ is a function of time $\eps$ as $\eps \to 0+$.
The growth, linear in $\eps$, is a new common qualitative feature shared with the Erdos-Renyi graph process and can be generalized to hypergraph processes with different max-allowed degree sequences. This is work in progress jointly with Lutz Warnke.
Title: Phase transitions in the 1-2 model
Abstract: A configuration in the 1-2 model is a subgraph of the hexagonal lattice, in which each vertex is incident to 1 or 2 edges. By assigning weights to configurations at each vertex, we can define a family of probability measures on the space of these configurations, such that the probability of a configuration is proportional to the product of weights of configurations at vertices. We study the phase transition of the model by investigating the probability measures with varying weights. We explicitly identify the critical weights, in the sense that the edge-edge correlation decays to 0 exponentially in the subcritical case, and converges to a non-zero constant in the supercritical case, under the limit measure obtained from torus approximation. These results are obtained by a novel measure-preserving correspondence between configurations in the 1-2 model and perfect matchings on a decorated graph, which appears to be a more efficient way to solve the model, compared to the holographic algorithm used by computer scientists to study the model. The major difficulty here is the absence of stochastic monotonicity.
Title: Fluctuations of the free energy of spherical Sherrington-Kirkpatrick model
Abstract: Consider the question of finding the maximum of a random polynomial defined on a closed manifold or a finite graph. Spherical Sherrington-Kirkpatrick (SSK) model is a finite temperature version of this question when the underlying space is a sphere. The free energy is the finite temperature version of the maximum value. The limit of the free energy as the dimension of the sphere becomes infinity is known by the works of Parisi, Cristanti, Sommers, and Talagrand. In this talk we consider the fluctuations when the polynomial is a symmetric quadratic function. We use a connection to random matrices and obtain limit theorems. This is a joint work with Ji Oon Li and Hao Wu.
Title: Conformally Invariant Paths and Loops
Abstract:
Abstract: There has been incredible progress in the last twenty years in the rigorous understanding of two-dimensional critical systems in statistical physics. I will give an overview with an emphasis on several related models: loop-erased random walk, spanning trees and corresponding loop soup; fractal paths and loops arising in critical systems (Schramm-Loewner evolution); the Gaussian free field and functions thereof (quantum gravity). I will also discuss some challenges for the future. This talk is intended for a general audience - it is not assumed that the audience is familiar with these terms.
Title: Phase transitions in some percolation models with long-range correlations on general graphs
Abstract:
Abstract: We consider two fundamental percolation models with long-range correlations on reasonably general and well-behaved transient graphs: The Gaussian free field and (the vacant set) of Random Interlacements. Both models have been the subject of intensive research during the last years and decades, on $\Z^d$ as well as on some more general graphs. We consider their percolation phase transition and investigate a couple of interesting properties of their critical parameters, in particular the existence of a phase transition.
This talk is based on joint works with A. Prevost (Koeln) and P.-F. Rodriguez (Los Angeles).
Title: Understanding rare events in models of statistical mechanics
Abstract: Statistical mechanics models are ubiquitous at the interface of probability theory, information theory, and inference problems in high dimensions. In this talk we will focus on sparse
graphs, and polymers on lattices; two canonical models in natural sciences. The study of large deviations is intimately related to the understanding of such models. We will consider the rare events that a sparse random network has an atypical number of certain local
structures and that a polymer in random media has atypical weight. Conditioning on such events can produce different, ranging from local to more global, geometric effects. We will discuss some such results obtained, relying on a variety of entropy theoretic, combinatorial, and analytic tools.
Title: Stationary Harmonic Measure and DLA in the Upper Half Plane
Abstract: In this talk, we introduce the stationary harmonic measure in the upper half plane. By bounding this measure, we are able to define both the discrete and continuous time diffusion limit aggregation (DLA) in the upper half plane with absorbing boundary conditions. We prove that for the continuous model the growth rate is bounded from above by $o(2+\epsilon)$. When time is discrete, we also prove a better upper bound of $o(2/3+\epsilon)$, on the maximum height of the aggregate at time $n$.
Title: Relating a classical planar map embedding algorithm to Liouville quantum gravity and SLE(16)
Abstract: In 1990, Walter Schnyder introduced a class of 3-spanning-tree decompositions of a simple triangulation to describe a combinatorially natural grid embedding algorithm for planar maps. It turns out that a uniformly sampled Schnyder-wood-decorated triangulation on n vertices converges as n tends to infinity to a random fractal surface, called a Liouville quantum gravity (LQG), together with a triple of intertwined fractal curves known as SLE(16). We will motivate this result by describing Schnyder’s algorithm and discussing some history of random planar map convergence results, and we will also introduce LQG and SLE and explain their role in the story.
Title: New perspectives on Mallows permutations
Abstract: I will discuss two projects concerning Mallows permutations, with Ander Holroyd, Tom Hutchcroft and Avi Levy. First, we relate the Mallows permutation to stable matchings, and percolation on bipartite graphs.
Second, we study the scaling limit of the cycles in the Mallows permutation, and relate it to diffusions and continuous trees.
Title: Dynamical freezing in a spin glass system with logarithmic correlations
Abstract: We consider a continuous time random walk on the two-dimensional discrete torus, whose motion is governed by the discrete Gaussian free field on the corresponding box acting as a potential. More precisely, at any vertex the walk waits an exponentially distributed time with mean given by the exponential of the field and then jumps to one of its neighbors, chosen uniformly at random. We prove that throughout the low-temperature regime and at in-equilibrium timescales, the process admits a scaling limit as a spatial K-process driven by a random trapping landscape, which is explicitly related to the limiting extremal process of the field. Alternatively, the limiting process is a supercritical Liouville Brownian motion with respect to the continuum Gaussian free field on the box. Joint work with Aser Cortines (University of Zurich) and Oren Louidor (Technion).
Title: The ant in high dimensional labyrinths
Abstract: One of the most famous open problem in random walks in random environments is to understand the behaviour of a simple random walk on a critical percolation cluster, a model known as the ant in the labyrinth. We will present new results on the scaling limit for the simple random walk on the critical branching random walk in high dimension which converges, after scaling, to the Brownian motion on the integrated super-brownian motion. In the light of lace expansion, we believe that the limiting behaviour of this model should be universal for simple random walks on critical structures in high dimension. In particular, recent progress show
that similar results hold for lattice trees.
Title: Stationary aggregation processes
Abstract: In this talk I'll introduce stationary versions of known aggregation models e.g., DLA, Hastings Levitov, IDLA and Eden. Using the additional symmetry and ergodic theory, one obtains new geometric insight on the aggregation processes.
Title: Gravitational allocation to uniform points on the sphere
Abstract: Given n uniform points on the surface of a two-dimensional sphere, how can we partition the sphere fairly among them? "Fairly" means that each region has the same area. It turns out that if the given points apply a two-dimensional gravity force to the rest of the sphere, then the basins of attraction for the resulting gradient flow yield such a partition-with exactly equal areas, no matter how the points are distributed. (See the cover of the AMS Notices at http://www.ams.org/publications/journals/notices/201705/rnoti-cvr1.pdf.) Our main result is that this partition minimizes, up to a bounded factor, the average distance between points in the same cell. I will also present an application to almost optimal matching of n uniform blue points to n uniform red points on the sphere, connecting to a classical result of Ajtai, Komlos and Tusnady (Combinatorica 1984). Joint work with Nina Holden and Alex Zhai.
Title: The strange geometry of high-dimensional random spanning forests.
Abstract: The uniform spanning forest (USF) in the lattice Z^d, first studied by Pemantle (Ann. Prob. 1991), is defined as a limit of uniform spanning trees in growing finite boxes. Although the USF is a limit of trees, it might not be connected- Indeed, Pemantle proved that the USF in Z^d is connected if and only if d<5. In later work with Benjamini, Kesten, and Schramm (Ann. Math 2004) we extended this result, and showed that the component structure of the USF undergoes a phase transition every 4 dimensions: For dimensions d between 5 and 8 there are infinitely many trees, but any two trees are adjacent; for d between 9 and 12 this fails, but for every two trees in the USF there is an intermediary tree, adjacent to each of the them. And this pattern continues, with the number of intermediary trees required increasing by 1 every 4 dimensions. In this talk, I will show that this is not the whole story, and for d>8 the USF geometry undergoes a qualitative change every time the dimension increases by 1. (Joint work with Tom Hutchcroft.)
Title: Random matrices, the Riemann zeta function and trees
Abstract: Fyodorov, Hiary & Keating have conjectured that the maximum of the characteristic polynomial of random unitary matrices behaves like extremes of log-correlated Gaussian fields. This allowed them to predict the typical size of local maxima of the Riemann zeta function along the critical axis. I will first explain the origins of this conjecture, and then outline the proof for the leading order of the maximum, for unitary matrices and the zeta function. This talk is based on joint works with Arguin, Belius, Radziwill and Soundararajan.
Title: Local limits of Random Sorting Networks
Abstract: A sorting network is a shortest path between 12..n and n..21 in the Cayley graph of the symmetric group spanned by swaps of adjacent letters. We will discuss the bulk
local limit of the swap process of uniformly random sorting networks and encounter
universal distributions of the random matrix theory, including the celebrated
Gaudin-Mehta law, which describes the energy level spacings in heavy nuclei.
Title: Geodesics in First-Passage Percolation
Abstract: First-passage percolation is a classical random growth model which comes from statistical physics. We will discuss recent results about the relationship between the limiting shape in first passage percolation and the structure of the infinite geodesics. This incudes a solution to the midpoint problem of Benjamini, Kalai and Schramm. This is joint work with Gerandy Brito and Daniel Ahlberg.
Title: Regenerative permutations: Mallows(q) and Riemann zeta function
Abstract: In this talk we discuss regenerative permutations on integers, with emphasis on two particular models: p-shifted and P-biased permutations. When p is the geometric distribution, the p-shifted permutations appear to be the limit of Mallows permutation model. We generalize and simplify previous work of Gnedin and Olshanski. The P-biased permutations are reminiscent of successive sampling in Bayesian statistics. Interestingly, some zeta formulas appear in the evaluation of renewal quantities of GEM-biased permutations. This is based on joint work with Jean-Jil Duchamps and Jim Pitman.
Title: Extremal metrics, eigenvalues, and graph separators
Title: Discrete conformal metrics and spectral geometry on distributional limits
Title: The SK model is FRSB at zero temperature
Abstract: In the early 80's, the physicist Giorgio Parisi wrote a series of ground breaking papers where he introduced the notion of replica symmetry breaking. His powerful insight allowed him to predict a solution for the SK model by breaking the symmetry of replicas infinitely many times.
In this talk, we will prove Parisi's prediction at zero temperature for the mixed p-spin model, a generalization of the SK model. We will show that at zero temperature the functional order parameter is full-step replica symmetry breaking (FRSB). We will also describe the importance of this result for the description of the energy landscape.
Based on recent works with Wei-Kuo Chen (U. of Minnesota) and Qiang Zeng (Northwestern U.).
Title: Limit theory for statistics of random geometric structures
Abstract: Questions arising in stochastic geometry and applied geometric probability are often understood in terms of the behavior of statistics of large random geometric structures.Such structures arise in diverse settings and include:
(i) Point processes of dependent points in R^d, including determinantal, permanental, and Gibbsian point sets, as well as the zeros of Gaussian analytic functions,
(ii) Simplicial complexes in topological data analysis,
(iii) Graphs on random vertex sets in Euclidean space,
(iv) Random polytopes generated by random data.
Global features of geometric structures are often expressible as a sum of local contributions. In general the local contributions have short range spatial interactions but complicated long range dependence. In this survey talk we review ``stabilization'' methods for establishing the limit theory for statistics of geometric structures. Stabilization provides conditions under which the behavior of a sum of local contributions is similar to that of a sum of independent identically distributed random variables.
Title: The Structure of Extreme Level Sets in Branching Brownian Motion
Abstract: We study the structure of extreme level sets of a standard one dimensional branching Brownian motion, namely the sets of particles whose height is within a fixed distance from the order of the global maximum. It is well known that such particles congregate at large times in clusters of order-one genealogical diameter around local maxima which form a Cox process in the limit. We add to these results by finding the asymptotic size of extreme level sets and the typical height and shape of those clusters which carry such level sets. We also find the right tail decay of the distribution of the distance between the two highest particles. These results confirm two conjectures of Brunet and Derrida (joint work with A. Cortines, O. Louidor).
Title: Branching capacity and critical branching random walks
Abstract: In this talk, I will introduce branching capacity for any finite subset of Z^d (d>=5). It turns out to be an important subject in the study of critical branching random walks. I will discuss its connections with critical branching random walks from the following three perspectives: 1) the hitting probability of a set by critical branching random walk; 2) branching recurrence and branching transience; 3) the local limit of critical branching random walk in torus.
Title: SLE loop measures
Abstract: An SLE loop measure is a $\sigma$-finite measure on the space of loops, which locally looks like a Schramm-Loewner evolution (SLE) curve. In this work, we use Minkowski content (i.e., natural parametrization) of SLE to construct several types of SLE$_\kappa$ loop measures for $\kappa\in(0,8)$. First, we construct rooted SLE$_\kappa$ loop measures in the Riemann sphere $\widehat{\mathbb C}$, which satisfy M\"obius covariance, conformal Markov property, reversibility, and space-time homogeneity, when the loop is parameterized by its $(1+\frac \kappa 8)$-dimensional Minkowski content. Second, by integrating rooted SLE$_\kappa$ loop measures, we construct the unrooted SLE$_\kappa$ loop measure in $\widehat{\mathbb C}$, which satisfies M\"obius invariance and reversibility. Third, we extend the SLE$_\kappa$ loop measures from $\widehat{\mathbb C}$ to subdomains of $\widehat{\mathbb C}$ and to Riemann surfaces using Brownian loop measures, and obtain conformal invariance or covariance of these measures. Finally, using a similar approach, we construct SLE$_\kappa$ bubble measures in simply/multiply connected domains rooted at a boundary point. The SLE$_\kappa$ loop measures for $\kappa\in(0,4]$ give examples of Malliavin-Kontsevich-Suhov loop measures for all $c\le 1$. The space-time homogeneity of rooted SLE$_\kappa$ loop measures in $\widehat{\mathbb C}$ answers a question raised by Greg Lawler.
Title: Extremal and local statistics for gradient field models
Abstract: We study the gradient field models with uniformly convex potential (also known as the Ginzburg-Landau field) in two dimension. This is a log-correlated (but generally non-Gaussian) random field that arises in different branches of mathematical physics. Previous results (Naddaf-Spencer, and Miller) were focused on the CLT for the linear functionals of the field. In this talk I will describe more precise results on the marginal distribution and the extreme values of the field. Based on joint works with David Belius and Ron Peled.
Title: Scaling limit of the directed polymer on Z^{2+1} in the critical window
Abstract: The directed polymer model on Z^{d+1} is the Gibbs transform of a directed random walk on Z^{d+1} in an i.i.d. random potential (disorder). It is known that the model undergoes a phase transition as the disorder strength varies, and disorder is relevant in d=1 and 2 in the sense that the presence of disorder, however weak, alters the qualitative behavior of the underlying random walk, with d=2 being the marginal case. For d=1, Alberts-Khanin-Quastel have shown that if the disorder strength tends to zero as a^{1/4} as the lattice spacing a tends to zero, then the partition functions converge to the solution of the Stochastic Heat Equation. We show that in the marginal dimension d=2, the partition functions admit non-trivial limits if the disorder strength scales as b/\sqrt{log 1/a}, with a transition at a critical point b_c. I will also discuss ongoing work in understanding the limit of the partition functions at b_c. Based on joint work with F. Caravenna and N. Zygouras.
Title: The Loewner Equation with Branching and the Continuum Random Tree
Abstract: In its most well-known form, the Loewner equation gives a correspondence between curves in the upper half-plane and continuous real functions (called the “driving function†for the equation). We consider the generalized Loewner equation, where the driving function has been replaced by a time-dependent real measure. In the first part of the talk, we investigate the delicate relationship between the driving measure and the generated hull, specifying a class of discrete random driving measures that generate hulls in the upper half-plane that are embeddings of trees. In the second part of the talk, we consider the scaling limit of these measures as the trees converge to the continuum random tree, with the goal of constructing an embedding of the CRT. We describe progress in this direction that has been obtained by analyzing the driving measures from an analytic standpoint, and we conclude by describing connections to the complex Burgers equation.
Title: Galton-Watson fixed points, tree automata, and interpretations
Abstract: Consider a set of trees such that a tree belongs to the set if and only if at least two of its root child subtrees do. One example is the set of trees that contain an infinite binary tree starting at the root. Another example is the empty set. Are there any other sets satisfying this property other than trivial modifications of these? I'll demonstrate that the answer is no, in the sense that any other such set of trees differs from one of these by a negligible set under a Galton-Watson measure on trees, resolving an open question of Joel Spencer's. This follows from a theorem that allows us to answer questions of this sort in general. All of this is part of a bigger project to understand the logic of Galton-Watson trees, which I'll tell you more about. Joint work with Moumanti Podder and Fiona Skerman.
Title: On loops of Brownian motion
Abstract: We provide a decomposition of the trace of the Brownian motion into a simple path and an independent Brownian soup of loops that intersect the simple path. More precisely, we prove that any subsequential scaling limit of the loop erased random walk is a simple path (a new result in three dimensions), which can be taken as the simple path of the decomposition. In three dimensions, we also prove that the Hausdorff dimension of any such subsequential scaling limit lies in (1, 5/3]. We conjecture that our decomposition characterizes uniquely the law of the simple path. If so, our results would give a new strategy to the existence of the scaling limit of the loop erased random walk and its rotational invariance.
Title: How round are the complementary components of planar Brownian motion?
Abstract:Consider a Brownian motion W in the complex plane started from 0 and run for time 1. Let A(1), A(2),... denote the bounded connected components of C-W([0,1]). Let R(i) (resp.\ r(i)) denote the out-radius (resp.\ in-radius) of A(i) for i \in N. Our main result is that E[\sum_i R(i)^2|\log R(i)|^\theta ]<\infty for any \theta<1. We also prove that \sum_i r(i)^2|\log r(i)|=\infty almost surely. These results have the interpretation that most of the components A(i) have a rather regular or round shape. Based on joint work with Serban Nacu, Yuval Peres, and Thomas S. Salisbury.
Friday, Mar 3: Wei Qian - ETH Zurich.
Title: Decomposition of Brownian loop-soup clusters
Abstract:We study the structure of Brownian loop-soup clusters in two dimensions.
The first part of the talk is based on joint-work with Wendelin Werner. Among other things, we obtain the following decomposition of the clusters with critical intensity: When one conditions a loop-soup cluster by its outer boundary $l$ (which is known to be an SLE4-type loop), then the union of all excursions away from $l$ by all the Brownian loops in the loop-soup that touch $l$ is distributed exactly like the union of all excursions of a Poisson point process of Brownian excursions in the domain enclosed by $l$.
In the second part of the talk, we condition a Brownian loop-soup cluster (of any intensity) on a portion $p$ of its boundary and show that the union of loops that touch $p$ satisfies the restriction property. This result implies that a phase transition occurs at c = 14/15 for the connectedness of the loops that touch $p$.
Title: Correlation inequalities for gradient fields and percolation
Abstract: We consider a class of massless gradient Gibbs measures, in dimension greater or equal to three, with uniformly convex potential (and non-convex perturbations thereof). A well-known example in this class is the Gaussian free field, which has received considerable attention in recent years. We derive a so-called decoupling inequality for these fields, which yields detailed information about their geometry, and the percolative and non-percolative phases of their level sets. An important aspect is the development of a suitable sprinkling technique, interesting in its own right, which we will discuss in some detail. Roughly speaking, it allows to dominate the strong correlations present in the model, and crucially relies on a particular representation of these correlations in terms of a random walk in a dynamic random environment, due to Helffer and Sjöstrand.
Title: Random planar geometry
Title: First-passage percolation in random planar lattices
Lecture 1: Taming infinities
Monday, October 24, 2016, 4:30pm–5:30pm, Ryerson 251
Abstract: Some physical and mathematical theories have the unfortunate feature that if one takes them at face value, many quantities of interest appear to be infinite! Various techniques, usually going under the common name of “renormalisation†have been developed over the years to address this, allowing mathematicians and physicists to tame these infinities. We will tip our toes into some of the mathematical aspects of these techniques and we will see how they have recently been used to make precise analytical statements about the solutions of some equations whose meaning was not even clear until now.
Lectures 2 and 3: The BPHZ theorem for stochastic PDEs
Tuesday, October 25, 2016, 4:30pm–5:30pm, Eckhart 202
Wednesday, October 26, 2016, 4pm–5pm, Eckhart 202
Abstract: The Bogoliubov-Parasiuk-Hepp-Zimmermann theorem is a cornerstone of perturbative quantum field theory: it provides a consistent way of "renormalising" the diverging integrals appearing there to turn them into bona fide distributions. Although the original article by Bogoliubov and Parasiuk goes back to the late 50s, it took about four decades for it to be fully understood. In the first lecture, we will formulate the BPHZ theorem as a purely analytic question and show how its solution arises very naturally from purely algebraic considerations. In the second lecture, we will show how a very similar structure arises in the context of singular stochastic PDEs and we will present some very recent progress on its understanding, both from the algebraic and the analytical point of view.
First Lecture : Modeling and Estimating Massive Networks: Overview
October 28, 4PM, Ryerson 251
Second Lecture: Limits and Stochastic Models for Sparse Massive Networks
October 31, 4PM, Eckhart 202
Third Lecture: Exchangeablity and Estimation of Sparse Massive Networks
November 1, 4PM, Eckhart 206
Title: The Borell-Ehrhard game
Abstract: A precise description of the convexity of Gaussian measures is provided by a remarkable Brunn-Minkowski type inequality due to Ehrhard and Borell. The delicate nature of this inequality has complicated efforts to develop more general geometric inequalities in Gauss space that mirror the rich family of results in the classical Brunn-Minkowski theory. In this talk, I will aim to shed some new light on Ehrhard's inequality by showing that it arises from a somewhat unexpected game-theoretic mechanism. This insight makes it possible to identify new results, such as an improved form of Barthe's reverse Brascamp-Lieb inequality in Gauss space. If time permits, I will also outline how probabilistic ideas enabled us (in work with Yair Shenfeld) to settle the equality cases in the Ehrhard-Borell inequalities.
Title: Percolative properties of Brownian interlacements and its vacant set
Abstract: In this talk, I will give a brief introduction to Brownian interlacements, and investigate various percolative properties regarding this model. Roughly speaking, Brownian interlacements can be described as a certain Poissonian cloud of doubly-infinite continuous trajectories in the d-dimensional Euclidean space, d greater or equal to 3, with the intensity measure governed by a level parameter. We are interested in both the interlacement set, which is an enlargement (“the sausagesâ€Â) of the union of the trace in the aforementioned cloud of trajectories, and the vacant set, which is the complement of the interlacement set. I will talk about the following results: 1) The interlacement set is “well-connectedâ€Â, i.e., any two “sausages†in d-dimensional Brownian interlacements, can be connected via no more than ceiling((d − 4)/2) intermediate sausages almost surely. 2) The vacant set undergoes a non-trivial percolation phase transition when the level parameter varies.
Title: Conformally invariant loop measures
Abstract: We will discuss several aspects of a conjecture by Kontsevich and Suhov regarding existence and uniqueness of a one parameter family of conformally invariant measures on simple loops (conjecturally related to the SLE family). The most natural case (zero central charge i.e. SLE parameter kappa=8/3) was understood in a paper of Werner predating the conjecture. In a work in progress, Dubédat and myself construct loop measures in the whole conjectural range of existence (i.e. parameters kappa for which SLE is a simple curve).
Title: Twin peaks
Abstract: I will discuss some questions and results on random labelings of graphs conditioned on having a small number of peaks (local maxima). The main open question is to estimate the distance between two peaks on a large discrete torus, assuming that the random labeling is conditioned on having exactly two peaks. Joint work with Sara Billey, Soumik Pal, Lerna Pehlivan and Bruce Sagan.
Title: Regularity structure theory and its applications
Abstract: Stochastic PDEs arise as important models in probability and mathematical physics. They are typically nonlinear, driven by very singular random forces. Due to lack of regularity it is typically very challenging to even interpret what one means by a solution. In this talk I will explain the solution theories for some of these equations, with a focus on the theory of regularity structures recently developed by Martin Hairer. As applications of these theories, one can make sense of the solutions to these stochastic PDEs, and once their solution theories are established various convergence or approximation problems can be tackled.
Title: First passage percolation on the exponential of two-dimensional branching random walk: subsequential scaling limit at high temperature
Abstract: Abstract: Let \{\eta_{N, v}: v\in V_N\} be a branching random walk in a two-dimensional box V_N of side length N, that is, a 4-ary BRW with Gaussian increments indexed by lattice points (with approximately log-correlated covariances). We study the first passage percolation metric where each vertex v is given a random weight of e^{\gamma \eta_{N, v}}. I will show that for sufficiently small but fixed \gamma>0, for any sequence of \{N_k\} there exists a subsequence along which the appropriately scaled FPP metric converges in the Gromov-Hausdorff sense to a random metric on the unit square in R^2. In addition, all possible (conjecturally unique) scaling limits are non-trivial and are continuous with respect to the Euclidean metric. Joint work with J. Ding.
Title: Delocalization and Universality of band matrices
Abstract: In this talk we introduce our new work on band matrices, whose eigenvectors and eigenvalues are widely believed to have the same asymptotic behaviors as those of Wigner matrices. We proved that this belief is true as long as the bandwidth is wide enough.
Title: Circle packing and uniform spanning forests of planar graphs
Abstract: The Koebe-Andreev-Thurston Circle Packing Theorem lets us draw planar graphs in a canonical way, so that the geometry of the drawing reveals analytic properties of the graph. Circle packing has proven particularly effective in the study of random walks on planar graphs, where it allows us to estimate various quantities in terms of their counterparts for Brownian motion in the plane. In this talk, I will introduce the theory of circle packing and discuss work with Asaf Nachmias in which we use circle packing to study uniform spanning forests of planar graphs, a probability model closely related to random walk. We prove that the free uniform spanning forest of any bounded degree, proper planar graph is connected almost surely, answering positively a question of Benjamini, Lyons, Peres and Schramm. Our proof is quantitative, and also shows that uniform spanning forests exhibit some of the same behaviour universally for all bounded degree transient triangulations, provided that one measures distances and areas in the triangulation using the hyperbolic geometry of its circle packing rather than with the usual graph metric and counting measure.
Title: An almost sure KPZ relation for SLE and Brownian motion
Abstract: I will discuss a KPZ-type formula which relates the Hausdorff dimension of any set associated with SLE, CLE, or related processes; and the Hausdorff dimension of a corresponding set associated with a correlated two-dimensional Brownian motion. In many cases, the dimension of the Brownian motion set is already known or easy to compute. This gives rise to new proofs of the dimensions of several sets associated with SLE, including the SLE curve; the double points and cut points of SLE; and the intersection of two flow lines of a Gaussian free field. The formula is based on the peanosphere construction of Duplantier, Miller, and Sheffield (2014), which encodes a Liouville quantum gravity (LQG) surface decorated with an independent space-filling SLE curve by means of a correlated two-dimensional Brownian motion. I will give a moderately detailed overview of this construction. Based on a joint work with Nina Holden and Jason Miller http://arxiv.org/abs/1512.01223.
Title: The Parisi variational problem
Abstract: The Parisi Variational Problem is a challenging non-local, strictly convex variational problem over the space of probability measures whose analysis is of great interest to the study of mean field spin glasses. In this talk, I present a conceptually simple approach to the study of this problem using techniques from PDEs, stochastic optimal control, and convex optimization. We begin with a new characterization of the minimizers of this problem whose origin lies in the first order optimality conditions for this functional. As a demonstration of the power of this approach, we study a prediction of de Almeida and Thouless regarding the validity of the 1 atomic anzatz. We generalize their conjecture to all mixed p-spin glasses and prove that their condition is correct in the entire temperature-external field plane except for a compact set whose phase is unknown at this level of generality. A key element of this analysis is a new class of estimates regarding gaussian integrals in the large noise limit called ``Dispersive Estimates of Gaussians’’ . This is joint work with Ian Tobasco (NYU Courant).
Title: Arm Exponents for SLE
Abstract: In the study of lattice models, arm exponents play an important role. In this talk, we first discuss the arm exponents for critical percolation, explain how they are derived and why they are important. Second, we introduce the arm exponents for chordal SLE and explain the application to the critical Ising and FK-Ising model. Finally, we give a brief idea on deriving these exponents and some related open questions.
Title: Convergence of naturally parametrized loop-erased random walk to the Schramm-Loewner evolution parametrized by Minkowski content
Abstract: The main goal of this talk is to explain the title. I will define the terms (type of convergence, naturally parametrized, loop-erased random walk, Schramm-Loewner evolution, Minkowski content) as well as the result. This is based on work with Fredrik Wiklund.
Title: Evolution of one-cells on a line
Abstract: We consider systems with the following description. At time zero, the real line is partitioned into intervals. The original partition, which may be random, evolves according to a deterministic rule whereby the interface between consecutive pair of cells move so that the larger cell grows and the smaller cell shrinks. When a cell shrinks to zero it disappears and the two bounding points coalesce. I will discuss one such system: a somewhat degenerate one-dimensional version of a two (and higher) dimensional mean-curvature flow model about which almost nothing rigorous is known. In joint work with Emanuel Lazar, we prove that the Poisson measure is invariant for this evolution, provided that space is rescaled exponentially. We do this by introducing the dual process (time-reversal). This process, unlike the forward process, contains some randomness and may be exactly analyzed. A number of questions remain open, such as uniqueness of trajectories, convergence to Poisson from other initial conditions, and stability under perturbation. Finally, I will discuss other one-dimensional models with similar descriptions about which even less is known.
Title: Correlation distillation in probability spaces
Abstract: Given a finite exchangeable collection of random variables in a probability space, the correlation distillation problem asks for the partition of the space into sets of a given measure as to maximize the probability that all random variables lie in the same set. This problem is closely related to isoperimetric problems and is motivated by applications in voting, theoretical computer science and information theory. In the talk I will survey some older and some recent results on correlation distillation. Many open problems will be presented.
Title: Dimensionality Reduction Via Sparse Matrices
Abstract: This talk will discuss sparse Johnson-Lindenstrauss transforms, i.e. sparse linear maps into much lower dimension which preserve the Euclidean geometry of a set of vectors. Both upper and lower bounds will be presented, as well as applications to certain domains such as numerical linear algebra and compressed sensing. Based on various joint works with Jean Bourgain, Sjoerd Dirksen, Daniel M. Kane, and Huy Le Nguyen.
Title: Speed of random walks on Cayley graphs of finitely generated groups
Abstract: In this walk I will discuss a new construction of a family of groups. We show that up to an absolute constant factor, any function $f$ satisfying $f(1)=1$, $f(n)/\sqrt{n}$, $n/f(n)$ both non-decreasing can be realized as speed function of simple random walk on some finitely generated group. In particular, it implies any number in [1/2,1] can be realized as the speed exponent of simple random walk on some group. The construction is very flexible and allows us to answer positively a recent conjecture of Gideon Amir regarding joint behavior of speed and entropy. We evaluate the Hilbert compression exponents of the groups under consideration. In particular, we show that for any $\alpha\in[2/3,1]$, there exists a 3-step solvable group with Hilbert compression exponent $\alpha$. It follows that there exists uncountably many pairwise non quasi-isometric finitely generated 3-step solvable groups. Joint work with Jeremie Brieussel.
Title: SPDE techniques for the random conductance model
Abstract: I will survey some of the recent work applying techniques from partial differential equations to the random conductance model on the lattice. This will include some work of mine with Armstrong and some work of Armstrong-Kuusi-Mourat and Gloria-Otto. There are now two approaches to obtaining optimal rates in stochastic homogenization in divergence form. The first obtains Green's function estimates by appealing to the Efron-Stein concentration inequality. The second uses regularity theory to localize the dependence of the solution on the coefficients. I will discuss both of these methods.
Title: Almost Sure Multifractal Spectrum of SLE
Abstract: 15 years ago B. Duplantier predicted the multifractal spectrum of Schramm Loewner Evolution (SLE), which encodes the fine structure of the harmonic measure of SLE curves. In this talk, I will report our recent rigorous derivation of this prediction. As a byproduct, we also confirm a conjecture of Beliaev and Smirnov for the a.s. bulk integral means spectrum of SLE. The proof uses various couplings of SLE and Gaussian free field, which are developed in the theory of imaginary geometry and Liouville quantum gravity. (Joint work with E. Gwynne and J. Miller.)
Title: The Hardy-Littlewood-Sobolev inequality via martingale transforms
Abstract: We outline a martingale proof of the classical Hardy-Littlewood-Sobolev (HLS) inequality which naturally extends to the setting of Markovian semigroups that have finite dimension in the sense of Varopoulos. The motivation for this approach comes from efforts to employ probabilistic techniques to study (extend) the sharp HLS inequality of E.H.Lieb.
Title: On multilevel Dyson Brownian motions
Abstract: I will discuss how Dyson Brownian motions describing the evolution of eigenvalues of random matrices can be extended to multilevel Dyson Brownian motions describing the evolution of eigenvalues of minors of random matrices. The construction is based on intertwining relations satisfied by the generators of Dyson Brownian motions of different dimensions. Such results allow to connect general beta random matrix theory to particle systems with local interactions, and to obtain novel results even in the case of classical GOE, GUE and GSE random matrix models. Based on joint work with Vadim Gorin.
Title: Isomorphism theorems for space-time random walks
Abstract: Loop measures have become important in the analysis of random walks and connected research in mathematical physics. Such measures go back to Symanzik in the late 1960s in the context of Euclidean field theory. We discuss loop measures on graphs with countable infinite different time horizons. These measures are connected to the cycle representation of partition functions in quantum systems (Boson systems). We derive corresponding Dynkin isomorphism theorems for space-time random walks and we prove for some specific models the onset of the so-called Bose-Einstein condensation.
Title: Discrete Fractal Dimensions and Large Scale Multifractals
Abstract: Ordinary fractal dimensions such as Hausdorff dimension and packing dimension are useful for analyzing the (microscopic) geometric structures of various thin sets and measures. For studying (macroscopic or global) fractal phenomena of discrete sets, Barlow and Taylor (1989, 1992) introduced the notions of discrete Hausdorff and packing dimensions. In this talk we present some recent results on macroscopic multifractal properties of random sets associated with the Ornstein-Uhlenbeck process and the mild solution of the parabolic Anderson model. (Joint work with Davar Khoshnevisan and Kunwoo Kim.)
Title: On multilevel Dyson Brownian motions.
Abstract: I will discuss how Dyson Brownian motions describing the evolution of eigenvalues of random matrices can be extended to multilevel Dyson Brownian motions describing the evolution of eigenvalues of minors of random matrices. The construction is based on intertwining relations satisfied by the generators of Dyson Brownian motions of different dimensions. Such results allow to connect general beta random matrix theory to particle systems with local interactions, and to obtain novel results even in the case of classical GOE, GUE and GSE random matrix models. Based on joint work with Vadim Gorin.
Title: A simple renormalization flow setup for FK-percolation models
Abstract: We will present a simple setup in which one can make sense of a renormalization flow for FK-percolation models in terms of a simple Markov process on a state-sace of discrete weighted graphs. We will describe how to formulate the universality conjectures in this framework (in terms of stationary measures for this Markov process), and how to prove this statement in the very special case of the two-dimensional uniform spanning tree (building on existing results on this model). This is based in part on joint work with Stéphane Benoist and Laure Dumaz.
Title: Dynamics on random regular graphs: Dyson Brownian motion and the Poisson free field
Abstract: : A single permutation, seen as union of disjoint cycles, represents a regular graph of degree two. Consider d many independent random permutations and superimpose their graph structures. It is a common model of a random regular (multi-) graph of degree 2d. Consider the problem of eigenvalue fluctuations of the adjacency matrix of such a graph. We consider the following dynamics. The ‘dimension’ of each permutation grows by coupled Chinese Restaurant Processes, while in ‘time’ each permutation evolves according to the random transposition Markov chain. Asymptotically in the size of the graph one observes a remarkable evolution of short cycles and linear eigenvalue statistics in dimension and time. We give a Poisson random surface description in dimension and time of the limiting cycle counts for every d. As d grows to infinity, these Poisson random surfaces converge to the Gaussian Free Field preserved in time by the Dyson Brownian motion. Part of this talk is based on a joint work with Tobias Johnson and the rest is based on a joint work with Shirshendu Ganguly. (Cambridge).
Title: Maxima of log-correlated Gaussian fields and of the Riemann Zeta function
Abstract: A recent conjecture of Fyodorov, Hiary & Keating states that the maxima of the Riemann Zeta function on a bounded interval of the critical line behave similarly to the maxima of a specific class of Gaussian fields, the so-called log-correlated Gaussian fields. These include important examples such as branching Brownian motion and the 2D Gaussian free field. In this talk, we will highlight the connections between the number theory problem and the probabilistic models. We will outline the proof of the conjecture in the case of a randomized model of the Zeta function. We will discuss possible approaches to the problem for the function itself. This is joint work with D. Belius (NYU) and A. Harper (Cambridge).
Title: Random matrices have simple spectrum
Abstract: A symmetric matrix has simple spectrum if all eigenvalues are different. Babai conjectured that random graphs have simple spectrum with probability tending to 1. Confirming this conjecture, we prove the simple spectrum property for a large class of random matrices. If time allows, we will discuss the harder problem of bounding the spacings between consecutive eigenvalues, with applications in mathematical physics, computer science, and numerical linear algebra. Several open questions will also be presented. Joint work with H. Nguyen (OSU) and T. Tao (UCLA).
Title: Exactly solvable mean-field monomer-dimers models
Abstract: A The seminar will introduce some mean-field models used to describe monomer-dimers systems. In particular the solution for the diluted case and the random impurity case will be shown and the absence of phase transition proved.
Title: A local central limit theorem for random representations of SU(3)
Abstract: The number p(n) of integer partitions of n is given approximately for large n by a famous asymptotic formula proved by Hardy and Ramanujan in 1918. This can be interpreted as a statement about the number of inequivalent representations of dimension n of the group SU(2). In this talk I will discuss my recent proof of an analogous result for the asymptotic number of n-dimensional representations of the group SU(3). A key step is to prove a local central limit theorem in a probabilistic model for random representations, which requires some ideas from the theory of modular forms. I will explain these ideas, as well as connections to a mysterious “Witten zeta function" associated with SU(3), and additional applications to understanding the limit shape of random n-dimensional representations of SU(3). No knowledge of representation theory will be assumed or needed.
Title: Effect of initial conditions on mixing for the Ising Model
Abstract: Recently, the ``information percolation'' framework was introduced as a way to obtain sharp estimates on mixing for the high temperature Ising model, and in particular, to establish cutoff in three dimensions up to criticality from a worst starting state. I will describe how this method can be used to understand the effect of different initial states on the mixing time, both random (''warm start'') and deterministic. Joint work with Allan Sly.
Title: Proportional Switching in FIFO Networks
Abstract: A central problem in queueing theory is the development of policies that efficiently allocate available resources. Many standard policies have a fixed capacity at individual sites, rather than the ability to allocate resources across sites. We discuss here the proportional switching policy, where the amount of service at different sites is dependent and the corresponding service vector is required to lie in a convex region. We also assume that packets are served in the FIFO (first-in, first-out) order. Past work on the stability of proportional switching networks has focused on networks with elementary routing structure (such as immediate departure after service at a site). Here, we consider the stability problem for general routing structures. The talk is based on joint work with B. D'Auria and N. Walton.
3:35-4:35 Friday, June 5th: Paul Jung -University of Alabama Birmingham.
Title: Levy Khintchine random matrices and the Poisson weighted infinite skeleton tree
Abstract: We study a class of Hermitian random matrices which includes Wigner matrices, heavy-tailed random matrices, and sparse random matrices such as adjacency matrices of Erdos-Renyi graphs with p=1/N. Our matrices have real entries which are i.i.d. up to symmetry. The distribution of entries depends on N, and we require sums of rows to converge in distribution; it is then well-known that the limit must be infinitely divisible. We show that a limiting empirical spectral distribution (LSD) exists, and via local weak convergence of associated graphs, the LSD corresponds to the spectral measure associated to the root of a graph which is formed by connecting infinitely many Poisson weighted infinite trees using a backbone structure of special edges. One example covered are matrices with i.i.d. entries having infinite second moments, but normalized to be in the Gaussian domain of attraction. In this case, the LSD is a semi-circle law.
Title: Recent results on the multispecies coalescent
Abstract: The multispecies coalescent is a variant of Kingman’s coalescent in which several populations are stitched together on a base tree. Increasingly, it plays an important role in phylogenetics where it can be used to model the joint evolution of a large number of genes across multiple species. Motivated by information-theoretic questions, I will present a recent probabilistic analysis of the multispecies coalescent which establishes fundamental limits on the inference of this model from molecular sequence data. No biology background is required. This is joint work with Gautam Dasarathy, Elchanan Mossel, Rob Nowak, and Mike Steel.
Title: Nash Equilibria for a Quadratic Voting Game
Abstract: Voters making a binary decision purchase votes from a centralized clearing house, paying the square of the number of votes purchased. The net payoff to an agent with utility u who purchases v votes is \Psi(S)u−v^2, where \Psi is an odd, monotone function taking values between -1 and +1 and S is the sum of all votes purchased by the n voters participating in the election. The utilities of the voters are assumed to arise by random sampling from a probability distribution F with compact support; each voter knows her own utility, but not those of the other voters, although she does know the sampling distribution F. Nash equilibria for this game are described.
Title: Brownian motion on spaces with varying dimension
Abstract: The model can be picturized as the random movement of an insect on the ground with a pole standing on it. That is, part of the state space has dimension 2, and the other part of the state space has dimension 1. We define such a process as a ``darning process'' in terms of Dirichlet form, because 2-dimensional Brownian motion does not hit any singleton. We show that the behavior of this process switches between 1-dimensional and 2-dimensional, which depends on both the time and the positions of the points. An open ongoing project will also be introduced: Can we approximate such a process by random walks? The main results of this talk are based on my joint work with Zhen-Qing Chen.
Title: Constant FacÂÂtor ApproxÂÂiÂÂmaÂÂtion for BalÂÂanced Cut in the PIE Model
Abstract: We proÂÂpose and study a new semi-random semi-adversarial model for BalÂÂanced Cut, a planted model with permutation-invariant ranÂÂdom edges (PIE). Our model is much more general than planted and stochastic modÂÂels conÂÂsidÂÂered preÂÂviÂÂously. ConÂÂsider a set of verÂÂtices V parÂÂtiÂÂtioned into two clusÂÂters L and R of equal size. Let G be an arbiÂÂtrary graph on V with no edges between L and R. Let E_random be a set of edges samÂÂpled from an arbiÂÂtrary permutation-invariant disÂÂtriÂÂbÂÂuÂÂtion (a disÂÂtriÂÂbÂÂuÂÂtion that is invariÂÂant under perÂÂmuÂÂtaÂÂtion of verÂÂtices in L and in R). Then we say that G + E_random is a graph with permutation-invariant ranÂÂdom edges. We present an approxÂÂiÂÂmaÂÂtion algoÂÂrithm for the BalÂÂanced Cut probÂÂlem that finds a balÂÂanced cut of cost O(|E_random|) + n polylog(n) in this model. In the regime when there are at least \Omega(n polylog(n)) random edges, this is a conÂÂstant facÂÂtor approxÂÂiÂÂmaÂÂtion with respect to the cost of the planted cut. Joint work with: Konstantin Makarychev and AravinÂÂdan Vijayaraghavan.
Title: Regularization under diffusion and Talagrand's convolution conjecture
Abstract: It is a well-known phenomenon that functions on Gaussian space become smoother under the Ornstein-Uhlenbeck semigroup. For instance, Nelson's hypercontractive inequality shows that if p > 1, then L^p functions are sent to L^q functions for some q > p. In 1989, Talagrand conjectured* that quantitative smoothing is achieved even for functions which are only L^1, in the sense that under the semigroup, such functions have tails that are strictly better than those predicted by Markov's inequality and preservation of mass. Ball, Barthe, Bednorz, Oleszkiewicz, and Wolff (2010) proved that this holds in fixed dimensions. We resolve Talagrand's conjecture conjecture positively (with no dimension dependence). The key insight is to study a subset of Gaussian space at various granularities by approaching it as "efficiently" as possible. To this end, we employ an Ito process that arose in the context of optimal control theory. Efficiency is measured by the average "work" required to couple the approach process to a Brownian motion. *Talagrand's full conjecture is for functions on the discrete cube. Here we address the Gaussian limiting case. This is joint work with Ronen Eldan.
Title: On the chemical distance in critical percolation
Abstract: In two-dimensional critical percolation, the works of Aizenman-Burchard and Kesten-Zhang imply that macroscopic distances inside percolation clusters are bounded below by a power of the Euclidean distance greater than 1+ε, for some positive ε. No more precise lower bound has been given so far. Conditional on the existence of an open crossing of a box of side length n, there is a distinguished open path which can be characterized in terms of arm exponents: the lowest open path crossing the box. This clearly gives an upper bound for the shortest path. The lowest crossing was shown by Zhang and Morrow to have volume n^4/3+o(1) on the triangular lattice. Following a question of Kesten and Zhang, we compare the length of shortest circuit in an annulus to that of the innermost circuit (defined analogously to the lowest crossing). I will explain how to show that the ratio of the expected length of the shortest circuit to the expected length of the innermost crossing tends to zero as the size of the annulus grows. Joint work with Jack Hanson and Michael Damron.
Title: High temperature limits for $(1+1)$-d directed polymer with heavy-tailed disorder.
Abstract: The directed polymer model at intermediate disorder regime was introduced by Alberts-Khanin-Quastel (2012). It was proved that at inverse temperature $\beta n^{-\gamma}$ with $\gamma=1/4$ the partition function, centered appropriately, converges in distribution and the limit is given in terms of the solution of the stochastic heat equation. This result was obtained under the assumption that the disorder variables posses exponential moments, but its universality was also conjectured under the assumption of six moments. We show that this conjecture is valid and we further extend it by exhibiting classes of different universal limiting behaviors in the case of less than six moments. We also explain the behavior of the scaling exponent for the log-partition function under different moment assumptions and values of $\gamma$. Based on joint work with Nikos Zygouras.
Title: Stochastic flows for Levy processes with Holder drifts
Abstract: In this talk I will present some new results on the following SDE in $R^d$: $$ dX_t=b(t, X_t)dt+dZ_t, \quad X_0=x, $$ where $Z$ is a Levy process. We show that for a large class of Levy processes $Z$ and Holder continuous drfit $b$, the SDE above has a unique strong solution for every starting point $x\in R^d$. Moreover, these strong solutions form a $C^1$-stochastic flow. In particular, we show that, when $Z$ is a symmetric $\alpha$-stable process with $\alpha\in (0, 1]$ and $b$ is $\beta$-Holder continuous with $\beta\in (1-\alpha/2, 1)$, the SDE above has a unique strong solution.
Title: Universality in spin glasses
Abstract: This talk is concerned about some universal properties of the Parisi solution in spin glass models. We will show universality of chaos phenomena and ultrametricity in the mixed p-spin model under mild moment assumptions on the environment. We will explain that the results also extend to quenched self-averaging of some physical observables in the mixed p-spin model as well as in different spin glass models including the Edwards-Anderson model and the random field Ising model.
Title: Displacement convexity of entropy and curvature in discrete settings
Abstract: Inspired by exciting developments in optimal transport and Riemannian geometry (due to the work of Lott-Villani and Sturm), several independent groups have formulated a (discrete) notion of curvature in graphs and finite Markov chains. I will describe some of these approaches briefly, and mention some related open problems of potential independent interest.
Title: Finitely Dependent Coloring on Z and other Graphs
Abstract: In 2008, Oded Schramm asked the following question: For what values of $k$ and $q$ does there exist a stationary, proper, $k-$dependent $q-$coloring of the integers? Schramm had a substantial amount of evidence, which I will describe, that convinced him that such a coloring does not exist for any values of $k$ and $q$. In fact, it turns out that such an object does exist for many values of $k$ and $q$. I will tell you exactly which ones work, and will describe colorings with these properties. No knowledge of advanced probability is needed to follow the lecture. There are several connections with combinatorics, but again, no specialized knowledge is needed. This is joint work with A. Holroyd.
Title: Stationary Eden Model on amenable groups
Abstract: We consider stationary versions of the Eden model, on a product of a Cayley graph G of an amenable group and positive integers. The process results in a collection of disjoint trees rooted at G, each of which consists of geodesic paths in a corresponding first passage percolation model on the product graph. Under weak assumptions on the weight distribution and by relying on ergodic theorems, we prove that almost surely all trees are finite. This generalizes certain known results on the two-type Richardson model, in particular of Deijfen and Haggstrom on the Euclidean lattice. This is a joint work with Eviatar Procaccia.
Title: Random tilings and Hurwitz numbers
Abstract: This talk is about random tilings of a special class of planar domains, which I like to call "sawtooth domains." Sawtooth domains have the special feature that their tilings are in bijective correspondence with Gelfand-Tsetlin patterns, aka semistandard Young tableaux. Consequently, many observables can be expressed in terms of special functions of representation-theoretic origin. In particular, the distribution of tiles of one type along a horizontal slice through a uniformly random tiling is encoded by the Harish-Chandra/Itzykson-Zuber integral, a familiar object from random matrix theory which also happens to be a generating function for a desymmetrized version of the Hurwitz numbers from enumerative algebraic geometry. I will explain how this fact allows one to prove that tiles along a slice fluctuate like the eigenvalues of a Gaussian random matrix.
Title: Lengths of Monotone Subsequences in a Mallows Permutation
Abstract: The longest increasing subsequence (LIS) of a uniformly random permutation is a well studied problem. Vershik-Kerov and Logan-Shepp first showed that asymptotically the typical length of the LIS is 2sqrt(n). This line of research culminated in the work of Baik-Deift-Johansson who related this length to the Tracy-Widom distribution. We study the length of the LIS and LDS of random permutations drawn from the Mallows measure, introduced by Mallows in connection with ranking problems in statistics. Under this measure, the probability of a permutation p in S_n is proportional to q^Inv(p) where q is a real parameter and Inv(p) is the number of inversions in p. We determine the typical order of magnitude of the LIS and LDS, large deviation bounds for these lengths and a law of large numbers for the LIS for various regimes of the parameter q. This is joint work with Ron Peled.
Title: Rate of convergence of the mean for sub-additive ergodic sequences
Abstract: For a subadditive ergodic sequence {X_{m,n}}, Kingman's theorem gives convergence for the terms X_{0,n}/n to some non-random number g. In this talk, I will discuss the convergence rate of the mean EX_{0,n}/n to g. This rate turns out to be related to the size of the random fluctuations of X_{0,n}; that is, the variance of X_{0,n}, and the main theorems I will present give a lower bound on the convergence rate in terms of a variance exponent. The main assumptions are that the sequence is not diffusive (the variance does not grow linearly) and that it has a weak dependence structure. Various examples, including first and last passage percolation, bin packing, and longest common subsequence fall into this class. This is joint work with Michael Damron and Jack Hanson.
Title: Hierarchical approximations to the Gaussian free field and fast simulation of Schramm-Loewner evolutions
Abstract: The Schramm--Loewner evolutions (SLE) are a family of stochastic processes which describe the scaling limits of curves which occur in two-dimensional critical statistical physics models. SLEs have had found great success in this task, greatly enhancing our understanding of the geometry of these curves. Despite this, it is rather difficult to produce large, high-fidelity simulations of the process due to the significant correlation between segments of the simulated curve. The standard simulation method works by discretizing the construction of SLE through the Loewner ODE which provides a quadratic time algorithm in the length of the curve. Recent work of Sheffield and Miller has provided an alternate description of SLE, where the curve generated is taken to be a flow line of the vector field obtained by exponentiating a Gaussian free field. In this talk, I will describe a new hierarchical method of approximately sampling a Gaussian free field, and show how this allows us to more efficiently simulate an SLE curve. Additionally, we will briefly discuss questions of the computational complexity of simulating SLE which arise naturally from this work.
Title: Limited choice and randomness in the evolution of networks
Abstract: The last few years have seen an explosion in network models describing the evolution of real world networks. In the context of math probability, one aspect which has seen an intense focus is the interplay between randomness and limited choice in the evolution of networks, ranging from the description of the emergence of the giant component, the new phenomenon of ``explosive percolation'' and power of two choices. I will describe ongoing work in understanding such dynamic network models, their connections to classical constructs such as the standard multiplicative coalescent and applications of these simple models in fitting retweet networks in Twitter.
Title: An example of hypoellipticity in infinite dimensions
Abstract: A collection of vector fields on a manifold satisfies H\"{o}rmander's condition if any two points are connected by a path whose tangent vectors only lie in the given directions. It is well-known that a diffusion which is allowed to travel only in these directions is smooth, in the sense that its transition probability measure is absolutely continuous with respect to the volume measure and has a strictly positive smooth density. Smoothness results of this kind in infinite dimensions are typically not known, the first obstruction being the lack of an infinite-dimensional volume measure. We will discuss recent results on a particular class of infinite-dimensional spaces, where we have shown that vector fields satisfying H\"{o}rmander's condition generate a diffusion which has a strictly positive smooth density with respect to an appropriate reference measure.
(Mathematics Colloquium, 2:00 -3:00 pm @ Eckhart 206 )
Title: Free probability and random matrices; from isomorphisms to universality
Abstract: Free probability is a probability theory for non-commutative variables introduced by Voiculescu about thirty years ago. It is equipped with a notion of freeness very similar to independence. It is a natural framework to study the limit of random matrices with size going to infinity. In this talk, we will discuss these connections and how they can be used to adapt ideas from classical probability theory to operator algebra and random matrices. We will in particular focus on how to adapt classical ideas on transport maps following Monge and Ampere to construct isomorphisms between algebras and prove universality in matrix models. This talk is based on joint works with F. Bekerman, Y. Dabrowski, A. Figalli and D. Shlyakhtenko.
Title: Strict Convexity of the Parisi Functional
Abstract: Spin glasses are magnetic systems exhibiting both quenched disorder and frustration, and have often been cited as examples of "complex systems." As mathematical objects, they provide several fascinating structures and conjectures. This talk will cover recent progress that shed more light in the mysterious and beautiful solution proposed 30 years ago by G. Parisi. We will focus on properties of the free energy of the famous Sherrington-Kirkpatrick model and we will explain a recent proof of the strict convexity of the Parisi functional. Based on a joint work with Wei-Kuo Chen.
(2:30-3:30) Elton P. Hsu - Northwestern University.
Title: Brownian Motion and Gradient Estimates of Positive Harmonic Functions
Abstract: Many gradient estimates in differential geometry can be naturally treated by stochastic methods involving Brownian motion on a Riemannian manifold. In this talk, we discuss Hamilton\'92s gradient estimate of bounding the gradient of the logarithm of a positive harmonic function in terms of its supremum from this point of view. We will see how naturally this gradient estimate follows from Ito\'92s formula and extend it to manifolds with boundary by considering reflecting Brownian motion. Furthermore, we will show that in fact Hamilton\'92s gradient estimate can be embedded as the terminal case of a family of gradient estimates which can be treated just as easily by the same stochastic method.
(4:00-5:00) Marek Biskup -UCLA.
Title: Isoperimetry for two dimensional supercritical percolation
Abstract: Isoperimetric problems have been around since ancient history. They play an important role in many parts of mathematics as well as sciences in general. Isoperimetric inequalities and the shape of isoperimetric sets are generally well understood in Euclidean or other "nice" settings but are still subject of research in random domains, graphs, manifolds, etc. In my talk I will address the isoperimetric problem for one example of a random setting: the unique infinite connected component of supercritical bond percolation on the square lattice. In particular, I will sketch a proof of the fact that, as the volume of a (properly defined) isoperimetric set tends to infinity, its asymptotic shape can be characterized by an isoperimetric problem in the plane with respect to a particular (continuum) norm. As an application I will conclude that that the anchored isoperimetric profile with respect to a given point as well as the Cheeger constant of the giant component in finite boxes scale to deterministic quantities. This settles a conjecture of Itai Benjamini for the plane. Based on joint work with O. Louidor, E. Procaccia and R. Rosenthal.
Title: Continuous spectra for sparse random graphs
Abstract: The limiting spectral distributions of many sparse random graph models are known to contain atoms. But a more interesting question is when they also have some continuous part. In this talk, I will give affirmative answer to this question for several widely studied models of random graphs including Erdos-Renyi random graph G(n,c/n) with c > 1, random graphs with certain degree distributions and supercritical bond percolation on Z^2. I will also present several open problems. This is joint work with Charles Bordenave and Balint Virag.
Title: Snowflakes, slot machines, Chinese dragons, and QLE
Abstract: What is the right way to think of a "random surface" or a "random planar graph"? How can one explain the dendritic patterns that appear in snowflakes, choral reefs, lightning bolts, and other physical systems, as well in as toy mathematical models inspired by these systems? How are these questions related to random walks and random fractal curves (in particular the famous SLE curves)? To begin to address these questions, I will introduce and explain the "quantum Loewner evolution", which is a family of growth processes closely related to SLE. I will explain. through pictures and animations and some discrete arguments, how QLE is defined and what role it might play in addressing the questions raised above. In a continuation of the talk on Friday afternoon (at the probability seminar), I will present a more analytic, continuum construction of QLE and discuss its relationship to the so-called Brownian map. Joint work with Jason Miller.
Title: Two-sided radial SLE and length-biased chordal SLE
Abstract: Models in statistical physics often give measures on self-avoiding paths. We can restrict such a measure to the paths that pass through a marked point, obtaining a "pinned measure". The aggregate of the pinned measures over all possible marked points is just the original measure biased by the path's length. Does the analogous result hold for SLE curves, which appear in the scaling limits of many such models at criticality? We show that it does: the aggregate of two-sided radial SLE is length-biased chordal SLE, where the path's length is measured in the natural parametrization.
Title: Planar growth models and conformal mapping
Abstract: Random, fractal-like growth can be seen in several places in nature. Several mathematical models based in one way or another on harmonic measure exist, but despite significant efforts little is known about these models. I will survey some of the models and problems, focusing in particular on constructions based on conformal maps. Towards the end I will discuss some recent joint work with Sola and Turner on one of these models.
Title: Tree Polymers Under Strong Disorder
Abstract: Tree polymers are simplifications of 1+1 dimensional lattice polymers made up of polygonal paths of a (nonrecombining) binary tree having random path probabilities. The path probabilities are (normalized) products of i.i.d. positive weights. As such, they reside in the more general framework of multiplicative cascades and branching random walk. The probability laws of these paths are of interest under weak and strong types of disorder. Some recent results, speculation and conjectures will be presented for this class of models under both weak and strong disorder conditions. This is based on various joint papers with Partha Dey, Torrey Johnson, or Stan Williams.
Wei Wu - Brown University. (3:35-4:35)
Title: Random fields from uniform spanning trees
Abstract: The uniform spanning tree (UST) is a fundamental combinatorial object. In two dimensions, using conformal invariance and planar duality, it is shown that the scaling limits of UST is given by one of the SLE path. We discuss the random field approach, and study the scaling limit of certain random fields coupled with USTs. This approach works on general graphs, and may help to understand the scaling limits of UST in higher dimensions. This talk is based on several joint works with Adrien Kassel, Richard Kenyon and Xin Sun.
Title: Intersections of SLE paths
Abstract: SLE curves are introduced by Oded Schramm as the candidate of the scaling limit of discrete models. In this talk, we first describe basic properties of SLE curves and their relation with discrete models. Then we summarize the Hausdorff dimension results related to SLE curves, in particular the new results about the dimension of cut points and double points. Third we introduce Imaginary Geometry, and from there give the idea of the proof of the dimension results.
Title: On Gaussian inequalities for product of functions
Abstract: Gaussian inequalities have played important roles in various scientific areas. In this talk, we will present simple algebraic criteria that yield sharp Holder types of inequalities for the product of functions of Gaussian random vectors with arbitrary covariance structure. As an application, we will explain how our results yield several famous inequalities in functional geometry, such as, the Brascamp-Lieb inequality, the sharp Young inequality, etc. This part of the talk is based on the recent joint work with N. Dafnis and G. Paouris. Along this direction, we will discuss a conjecture on the convexity of the Parisi functional arising from the study of the Sherrington-Kirkpatrick model in spin glass.
Title: Conformal invariance of the Green's function for
loop-erased random walk
Abstract: The planar loop-erased random walk (LERW) is obtained from the usual random walk by erasing loops. The LERW is related to a number of other models such as the uniform spanning tree. We consider a fixed simply connected domain in C containing the origin, and two distinct boundary points a and b. For a fixed lattice spacing, we consider the probability that a LERW goes from a to b goes through an edge containing the origin. We show that the normalized limit of this probability goes to a conformally covariant quantity, the Green's function for the Schramm-Loewner evolution. This is joint work with Christian Benes and Fredrik Viklund .
Title: Random walks on planar graphs via circle packings
Abstract: I will describe two results concerning random walks on planar graphs and the connections with Koebe's circle packing theorem (which I will not assume any knowledge of): 1. A bounded degree planar triangulation is recurrent if an only if the set of accumulation points of its circle packing is a polar set (that is, has zero logarithmic capacity). This extends a result of He and Schramm who proved recurrence (transience) when the set of accumulation points is empty (a closed Jordan curve). Joint work with Ori Gurel-Gurevich and Juan Souto. 2. The Poisson boundary (the space of bounded harmonic functions) of a transient bounded degree triangulation of the plane is characterized by the topological boundary obtained by circle packing the graph in the unit disk. In other words, any bounded harmonic function on the graph is the harmonic extension of some measurable function on the boundary of the unit disc. Joint work with Omer Angel, Martin Barlow and Ori Gurel-Gurevich.
Title: A Two-Sided Estimate for the Gaussian Noise Stability Deficit
Abstract: The Gaussian Noise Stability of a set A in Euclidean space is the probability that for a Gaussian vector X conditioned to be in A, a small Gaussian perturbation of X will also be in A. Borell's celebrated inequality states that a half-space maximizes the noise stability among all possible sets having the same Gaussian measure. We present a novel short proof of this inequality, based on stochastic calculus. Moreover, we prove an almost tight, two-sided, dimension-free robustness estimate for this inequality: We show that the deficit between the noise stability of a set A and an equally probable half-space H can be controlled by a function of the distance between the corresponding centroids. As a consequence, we prove a conjecture of Mossel and Neeman, who used the total-variation distance as a metric.
Title: Asymptotic behavior of log-concave probability measures
Abstract: A probability measure $\mu$ in ${\mathbb R}^n$ is called log-concave if $\mu\big(\lambda A + (1-\lambda) B\big) \geq \mu(A)^\lambda\,\mu(B)^{1-\lambda}$, for every $\lambda\in[0,1]$ and every $A,B$ Borel subsets of ${\mathbb R}^n$. Two basic examples are the uniform measure restricted to a convex body in ${\mathbb R}^n$ with volume $1$ (Brunn-Minkowski inequality) and the normal Gaussian measure in ${\mathbb R}^n$. We are studying the asymptotic behavior of some random geometric quantities such as the volume and the radius of a random polytope generated by sampling with respect to a log-concave probability measure. We will show that asymptotically ( as the dimension $n$ goes to infinity), they behave like if we had sampled with respect to the Gaussian measure.
Title: Performance of the Metropolis algorithm on a disordered tree: the Einstein relation.
Abstract: Consider a d-ary rooted tree (d>2) where each edge e is assigned an i.i.d. (bounded) random variable X(e) of negative mean. Assign to each vertex v the sum S(v) of X(e) over all edges connecting v to the root, and assume that the maximum S_n* of S(v) over all vertices v at distance n from the root tends to infinity (necessarily, linearly) as n tends to infinity. We analyze the Metropolis algorithm on the tree and show that under these assumptions there always exists a temperature of the algorithm so that it achieves a linear (positive) growth rate in linear time. This confirms a conjecture of Aldous (Algorithmica, 22(4):388-412, 1998). The proof is obtained by establishing an Einstein relation for the Metropolis algorithm on the tree. Joint work with Pascal Maillard.
Title: Persistence Probabilities.
Abstract: Persistence probabilities concern how likely it is that a stochastic process has a long excursion above fixed level and of what are the relevant scenarios for this behavior. Power law decay is expected in many cases of physical significance and the issue is to determine its power exponent parameter. I will survey recent progress in this direction (jointly with Sumit Mukherjee), dealing with stationary Gaussian processes that arise from random algebraic polynomials of independent coefficients and from the solution to heat equation initiated by white noise. If time permits, I will also discuss the relation to joint works with Jian Ding and Fuchang Gao, about persistence for iterated partial sums and other auto-regressive sequences, and to the work of Sakagawa on persistence probabilities for the height of certain dynamical random interface models.
Title: Statistics on Hilbert's Sixteenth Problem
Abstract: The first part of Hilbert's sixteenth problem concerns real algebraic geometry: We are asked to study the number and possible arrangements of the connected components of a real algebraic curve (or hypersurface). I will describe a probabilistic approach to studying the topology, volume, and arrangement of the zero set (in real projective space) of a random homogeneous polynomial. The outcome depends on the definition of "random". A popular Gaussian ensemble uses monomials as a basis, but we will favor eigenfunctions on the sphere (spherical harmonics) as a basis. As we will see, this "random wave" model produces a high expected number of components (a fraction of the Harnack bound that was an inspiration for Hilbert's sixteenth problem). This is joint work with Antonio Lerario.
Title: A Dynamic Graph Model of Barter Exchanges
Abstract: Motivated by barter exchanges, we study average waiting time in a dynamic random graph model. A node arrives at each time step. A directed edge is formed independently with probability p with each node currently in the system. If a cycle is formed, of length no more than 3, then that cycle of nodes is removed immediately. We show that the average waiting time for > a node scales as 1/p^{3/2} for small p, for this policy. Moreover, we prove that we cannot achieve better delay scaling by batching. Our results through new light on the operation of kidney exchange programs. The insight offered by our analysis is that the beneï¬Ât of waiting for additional incompatible patient-donor pairs to arrive (batching) into kidney exchange clearinghouses is not substantial and is outweighed by the cost of waiting. Joint work with Ross Anderson, Itai Ashlagi and David Gamarnik.
Title: Epsilon-biased sets, the Legendre symbol, and getting by with a few random bits
Abstract: Subsets of F_2^n that are p-biased, meaning that the parity of any set of bits is even or odd with probability close to 1/2, are useful tools in derandomization. They also correspond to optimal error-correcting codes,i.e. meeting the Gilbert-Varshamov bound, with distance close to n/2. A simple randomized construction shows that such sets exist of size O(n/p^2); recently, Ben-Aroya and Ta-Shma gave a deterministic construction of size O((n/p^2)^(5/4)). I will review deterministic constructions of Alon, Goldreich, Haastad, and Peralta of sets of size O(n/p^3) and O(n^2/p^2), and discuss the delightful pseudorandom properties of the Legendre symbol along the way. Then, rather than derandomizing these sets completely in exchange for making them larger, we will try moving in a different direction on the size-randomness plane, constructing sets of optimal size O(n/p^2) with as few random bits as possible. The naive randomized construction requires O(n^2/p^2) random bits. I will show that this can be reduced to O(n log(n/p)) random bits. Like Alon et al., our construction uses the Legendre symbol and Weil sums, but in a different way to control high moments of the bias. I'll end by saying a few words about Ramsey graphs and random polynomials. This is joint work with Alex Russell.
Title: Quantum spin systems and graphical representations
Abstract: Quantum spin systems are mathematical models for magnetism. But the quantum nature is a difficulty. For some models there are graphical representations, which relate to interacting particle processes (with some changes). I will discuss one application done jointly with Nick Crawford and Stephen Ng, called emptiness formation probability where this approach works.
Title: Delocalization of eigenvectors of random matrices
Abstract: Eigenvectors of random matrices are much less studied than eigenvalues, despite their importance. The simplest question is whether the eigenvectors are delocalized, i.e. all of their coordinates are as small as can be, of order n^{-1/2}. Even this simple looking problem has been open until very recently. Currently there are two approaches to delocalization - spectral (via local eigenvalue statistics) and geometric (via high dimensional probability). This talk will explain these approaches and popularize related open problems. Based on joint work with Mark Rudelson (Michigan).
Title: Multiple Phase Transitions for long range first-passage percolation on lattices
Abstract: Given a graph G with non-negative edge weights, the passage time of a path is the sum of weights of the edges in the path, and the first-passage time to reach u from v is the minimum passage time of a path joining them. We consider a long range first-passage model on Z^d in which, the weight w(x,y) of the edge joining x and y has exponential distribution with mean |x-y|^a for some fixed a > 0, and the edge weights are independent. We analyze the growth of the set of vertices reachable from the origin within time t, and show that there are four different growth regimes depending on the value of a. Joint work with Partha Dey.
Title: Statistics on Hilbert's Sixteenth Problem
Abstract: The first part of Hilbert's sixteenth problem concerns real algebraic geometry: We are asked to study the number and possible arrangements of the connected components of a real algebraic curve (or hypersurface). I will describe a probabilistic approach to studying the topology, volume, and arrangement of the zero set (in real projective space) of a random homogeneous polynomial. The outcome depends on the definition of "random". A popular Gaussian ensemble uses monomials as a basis, but we will favor eigenfunctions on the sphere (spherical harmonics) as a basis. As we will see, this "random wave" model produces a high expected number of components (a fraction of the Harnack bound that was an inspiration for Hilbert's sixteenth problem). This is joint work with Antonio Lerario.
Title: On the monotonicity of the speed of biaised random walk on a Galton-Watson tree without leaves.
Abstract: We will present different results related to the speed of biased random walks in random environments. Our focus will be on a recent paper by Ben Arous, Fribergh and Sidoravicius proving that the speed of the biased random walk on a Galton-Watson tree without leaves is increasing for high biases. This partially solves a question asked by Lyons, Pemantle and Peres.
Title: Search Games, The Cauchy process and Optimal Kakeya Sets
Abstract: A planar set that contains a unit segment in every direction is called a Kakeya set. These sets have been studied intensively in geometric measure theory and harmonic analysis since the work of Besicovich (1928); we find a new connection to game theory and probability via a search game first analyzed by Adler et al (2003). A hunter and a rabbit move on the n-vertex cycle without seeing each other. At each step, the hunter moves to a neighboring vertex or stays in place, while the rabbit is free to jump to any node. Thus they are engaged in a zero sum game, where the payoff is the capture time. The known optimal randomized strategies for hunter and rabbit achieve expected capture time of order n log n. We show that every rabbit strategy yields a Kakeya set; the optimal rabbit strategy is based on a discretized Cauchy random walk, and it yields a Kakeya set K consisting of 4n triangles, that has minimal area among such sets (the area of K is of order 1/log(n)). Passing to the scaling limit yields a simple construction of a random Kakeya set with zero area from two Brownian motions. (Joint work with Y. Babichenko, R. Peretz, P. Sousi and P. Winkler).
Tutorial Seminar: What is the mixing time for random walk on a graph?
Abstract: Consider a simple random walk on a finite graph. The mixing time is the time it takes the walk to reach a position that is approximately independent of the starting point; it has been studied intensively by combinatorialists, computer scientists and probabilists; the mixing time arises in statistical physics as well. Applications of mixing times range from random sampling and card shuffling, to understanding convergence to equilibrium in the Ising model. It is closely related to expansion and eigenvalues. Besides introducing this topic, I will also describe the open problem of understanding which random walks exhibit "cutoff", a sharp transition to stationarity first discovered by Diaconis, Shashahani and Aldous in the early 1980s but still mysterious.
Title: Convergence to equilibrium for nonreversible diffusions.
Abstract: The problem of convergence to equilibrium for diffusion processes is of theoretical as well as applied interest, for example in nonequilibrium statistical mechanics and in statistics, in particular in the study of Markov Chain Monte Carlo (MCMC) algorithms. Powerful techniques from analysis and PDEs, such as spectral theory and functional inequalities (e.g. logarithmic Sobolev inequalities) can be used in order to study convergence to equilibrium. Quite often, the diffusion processes that appear in applications are degenerate (in the sense that noise acts directly to only some of the degrees of freedom of the system) and/or nonreversible. The study of convergence to equilibrium for such systems requires the study of non-selfadjoint, possibly non-uniformly elliptic, second order differential operators. In this talk we show how the recently developed theory of hypocoercivity can be used to prove exponentially fast convergence to equilibrium for such diffusion processes. Furthermore, we will show how the addition of a nonreversible perturbation to a reversible diffusion can speed up convergence to equilibrium. This is joint work with M. Ottobre, K. Pravda-Starov, T. Lelievre and F. Nier.
This is a special event. Billingsley Lectures on Probability in honor of Professor Billingsley.
Title: Random Walk with Reinforcement
Abstract: Picture a triangle, with vertices labeled A, B, C. A random walker starts at A and chooses a random nearest neighbor. At each stage, the walker adds 1 to the weight of each crossed edge and chooses the next step with probability proportional to the current edge weights. The question is 'what happens?'. This simple problem leads into interesting corners: to Bayesian analysis of the transition mechanism of Markov chains (and protein folding) and to the hyperbolic sigma model of statistical physics. Work of (and with) Billingsley, Baccalado, Freedman, Tarres, and Sabot will be reviewed.
Title: Exchangeable random measures
Abstract: Classical theorems of de Finetti, Aldous-Hoover and Kallenberg describe the structure of exchangeable probability measures on spaces of sequences or arrays. Similarly, one can add an extra layer of randomness, and ask after exchangeable random measures on these spaces. It turns out that those classical theorems, coupled with an abstract version of the `replica trick' from statistical physics, give a structure theorem for these random measures also. This leads to a new proof of the Dovbysh-Sudakov Theorem describing exchangeable positive semi-definite matrices.
Title: Maximum independent sets in random d-regular graphs
Abstract: Satisfaction and optimization problems subject to random constraints are a well-studied area in the theory of computation. These problems also arise naturally in combinatorics, in the study of sparse random graphs. While the values of limiting thresholds have been conjectured for many such models, few have been rigorously established. In this context we study the size of maximum independent sets in random d-regular graphs. We show that for d exceeding a constant d(0), there exist explicit constants A, C depending on d such that the maximum size has constant fluctuations around A*n-C*(log n) establishing the one-step replica symmetry breaking heuristics developed by statistical physicists. As an application of our method we also prove an explicit satisfiability threshold in random regular k-NAE-SAT. This is joint work with Jian Ding and Allan Sly.
Title: Scaling limit of the abelian sandpile
Abstract: Which functions of two real variables can be expressed as limits of superharmonic functions from (1/n)Z2 to (1/n2)Z? I'll discuss joint work with Wesley Pegden and Charles Smart on the case of quadratic functions, where this question has a surprising and beautiful answer: the maximal such quadratics are classified by the circles in a certain Apollonian circle packing of the plane. I'll also explain where the question came from (the title is a hint!).
Title: The relaxation of a family of broken bond crystal surface models
Abstract: We study the continuum limit of a family of kinetic Monte Carlo models of crystal surface relaxation that includes both the solid-on-solid and discrete Gaussian models. With computational experiments and theoretical arguments we are able to derive several partial differential equation (PDE) limits identified (or nearly identified) in previous studies and to clarify the correct choice of surface tension appearing in the PDE and the correct scaling regime giving rise to each PDE. We also provide preliminary computational investigations of a number of interesting qualitative features of the large scale behavior of the models.
Title: Random polymers and last passage percolation: variational formulas, Busemann functions, geodesics, and other stories
Abstract: We give variational formulas for random polymer models, both in the positive- and zero-temperature cases. We solve these formulas in the oriented two-dimensional zero-temperature case. The solution comes via proving almost-sure existence of the so-called Busemann functions. We then use these results to prove existence, uniqueness, and coalescence of semi-infinite directional geodesics, for exposed points of differentiability of the limiting shape function.
Title: The scaling limit of simple triangulations and quadrangulations
Abstract: A graph is simple if it contains no loops or multiple edges. We establish Gromov--Hausdorff convergence of large uniformly random simple triangulations and quadrangulations to the Brownian map, answering a question of Le Gall (2011). In proving the preceding fact, we introduce a labelling function for the vertices of the triangulation. Under this labelling, distances to a distinguished point are essentially given by vertex labels, with an error given by the winding number of an associated closed loop in the map. The appearance of a winding number suggests that a discrete complex-analytic approach to the study of random triangulations may lead to further discoveries. Joint work with Marie Albenque.
Title Law of the extremes for the two-dimensional discrete Gaussian Free Field
Abstract: A two-dimensional discrete Gaussian Free Field (DGFF) is a centered Gaussian process over a finite subset (say, a square) of the square lattice with covariance given by the Green function of the simple random walk killed upon exit from this set. Recently, much effort has gone to the study of the concentration properties and tail estimates for the maximum of DGFF. In my talk I will address the limiting extreme-order statistics of DGFF as the square-size tends to infinity. In particular, I will show that for any sequence of squares along which the centered maximum converges in law, the (centered) extreme process converges in law to a randomly-shifted Gumbel Poisson point process which is decorated, independently around each point, by a random collection of auxiliary points. If there is any time left, I will review what we know and/or believe about the law of the random shift. This talk is based on joint work with Oren Louidor (UCLA).
Title: The Virasoro algebra and discrete Gaussian free field
Abstract: The Virasoro algebra is an infinite dimensional Lie algebra that plays an important role in the Conformal Field Theory (CFT) methods employed by physicists to describe and study conformally invariant scaling limits of planar critical lattice models from statistical physics. Despite much progress in the last decade, it seems fair to say that from a mathematical perspective many aspects of the connections between discrete model and continuum limit CFT remain somewhat mysterious. In the talk I will discuss recent joint work with C. Hongler and K. Kytola concerning the discrete Gaussian free field on a square grid. I will explain how for this model discrete complex analysis can be used to construct explicit (exact) representations of the Virasoro algebra of central charge 1 directly on the discrete level.
Title: Markov type and the multi-scale geometry of metric spaces
Abstract: The behavior of random walks on metric spaces can sometimes be understood by embedding such a walk into a nicer space (e.g. a Hilbert space) where the geometry is more readily approachable. This beautiful theme has seen a number of geometric and probabilistic applications. We offer a new twist on this study by showing that one can employ mappings that are significantly weaker than bi-Lipschitz. This is used to answer questions of Naor, Peres, Schramm, and Sheffield (2004) by proving that planar graph metrics and doubling metrics have Markov type 2. The main new technical idea is that martingales are significantly worse at aiming than one might at first expect. Joint work with Jian Ding and Yuval Peres.
Title: The Renormalization Group for Disordered Systems
Abstract: We investigate the Renormalization Group (RG) approach in finite- dimensional glassy systems, whose critical features are still not well-established, or simply unknown. We focus on spin and structural-glass models built on hierarchical lattices, which are the simplest non-mean-field systems where the RG framework emerges in a natural way. The resulting critical properties shed light on the critical behavior of spin and structural glasses beyond mean field, and suggest future directions for understanding the criticality of more realistic glassy systems.
Title: Geodesics and Direction in 2d First-Passage Percolation
Abstract: I will discuss geodesics in first-passage percolation, a model for fluid flow in a random medium. There are numerous conjectures about the existence, coalescence, and asymptotic direction of infinite geodesics under the model's random metric. C. Newman and collaborators have proved some of these under strong assumptions. I will explain recent results with Michael Damron which develop a framework for addressing these questions; this framework allows us to prove versions of Newman's results under minimal assumptions.
Title: Gaussian Free Field fluctuations for general-beta random matrix ensembles.
Abstract: It is now known that the asymptotic fluctuations of the height function of uniformly random lozenge tilings of planar domains (equivalently, stepped surfaces in 3d space) are governed by the Gaussian Free Field (GFF), which is a 2d analogue of the Brownian motion. On the other hand, in certain limit regimes such tilings converge to various random matrix ensembles corresponding to beta=2. This makes one wonder whether GFF should also somehow arise in general-beta random matrix ensembles. I will explain that this is indeed true and the asymptotics of fluctuations of classical general-beta random matrix ensembles is governed by GFF. This is joint work with A.Borodin.
Title: About heavy tailed random matrices.
Abstract:We investigate the behaviour of matrices which do not
belong to the universality class of Wigner matrices because their entries
have heavy tails.
Title: Chaos problem in mean field spin glasses
Abstract: The main objective in spin glasses from the physical perspective is to understand the strange magnetic properties of certain alloys. Yet the models invented to explain the observed phenomena are also of a rather fundamental nature in mathematics. In this talk we will first introduce the famous Sherrington-Kirkpatrick model as well as some known results about this model such as the Parisi formula and the limiting behavior of the Gibbs measure. Next, we will discuss the problems of chaos in the mixed p-spin models and present mathematically rigorous results including disorder, external field, and temperature chaos.
Abstract: This seminar was canceled. It will be rescheduled.
Title: The 2D SOS Model
Abstract: We present new results on the (2+1)-dimensional Solid-On-Solid model at low temperatures. Bricmont, El-Mellouki and Froelich (1986) showed that in the presence of a floor there is an entropic repulsion phenomenon, lifting the surface to a height which is logarithmic in the side of the box. We refine this and establish the typical height of the SOS surface is precisely the floor of [1/(4\beta)\log n], where n is the side-length of the box and \beta is the inverse-temperature. We determine the asymptotic shape of the top plateau and show that its boundary fluctuation are n^{1/3+o(1)}. Based on joint works with Pietro Caputo, Eyal Lubetzky, Fabio Martinelli and Fabio Toninelli.
Title: Spiking the random matrix hard edge.
Abstract: The largest eigenvalue of a finite rank perturbation of a random hermitian matrix is known to exhibit a phase transition (in the infinite dimensional limit). If the perturbation is small one sees the famous Tracy-Widom law, while a large perturbation results in a Gaussian fluctuation. In between there exists is a scaling window about a critical perturbation value leading to a separate family of limit laws. This basic discovery is due to Baik, Ben Arous, and Peche. More recently Bloemendal and Virag have shown this picture persists in the context of the general beta ensembles, giving new formulations of the critical limit laws . Yet another route, explained here, is to go through the random matrix hard edge, perturbing the smallest eigenvalues in the sample covariance set-up. A limiting procedure then recovers all the alluded to distributions. (Joint work with Jose Ramirez.)
Title: Directed polymers and the stochastic heat equation
Abstract: We show how some properties of the solutions of the Stochastic Heat Equation (SHE) can be derived from directed polymers in random environment. In particular, we show: * A new proof of the positivity of the solutions of the SHE * Improved bounds on the negative moments of the SHE * Results on the fluctuations of the log of the SHE in equilibrium, namely, the Cole-Hopf solution of the KPZ equation (if time allows).
Title: Second-order Boltzmann-Gibbs principle and applications
Abstract: The celebrated Botzmann-Gibbs principle introduced by Rost in the 80's roughly says the following. For stochastic systems with one or more conservation laws, fluctuations of the non-conserved quantities are faster than fluctuations of the conserved quantities. Therefore, in the right space-time window, the space-time fluctuations of a given observable are asymptotically equivalent to a linear functional of the conserved quantities. In one dimension, we prove two generalizations of this principle: a non-linear (or second-order) and a local version of it. This result opens a way to show convergence of fluctuations for non-linear models, like the ones on the fashionable KPZ universality class. As a corollary, we prove new convergence results for various observables of the asymmetric exclusion process, given in terms of solutions of the KPZ equation. Joint work with Patricia Gonçalves.
Title: SLE curves and natural parametrization
Title: Robust Gaussian noise stability
Abstract: Given two Gaussian vectors that are positively correlated, what is the probability that they both land in some fixed set A? Borell proved that this probability is maximized (over sets A with a given volume) when A is a half-space. We will give a new and simple proof of this fact, which also gives some stronger results. In particular, we can show that half-spaces uniquely maximize the probability above, and that sets which almost maximize this probability must be close to half-spaces.
Title: Spiking the random matrix hard edge.
Abstract: The largest eigenvalue of a finite rank perturbation of a random hermitian matrix is known to exhibit a phase transition (in the infinite dimensional limit). If the perturbation is small one sees the famous Tracy-Widom law, while a large perturbation results in a Gaussian fluctuation. In between there exists is a scaling window about a critical perturbation value leading to a separate family of limit laws. This basic discovery is due to Baik, Ben Arous, and Peche. More recently Bloemendal and Virag have shown this picture persists in the context of the general beta ensembles, giving new formulations of the critical limit laws . Yet another route, explained here, is to go through the random matrix hard edge, perturbing the smallest eigenvalues in the sample covariance set-up. A limiting procedure then recovers all the alluded to distributions. (Joint work with Jose Ramirez.)
Title: Extreme values for random processes of tree structures
Abstract: The main theme of this talk is that studying implicit tree structures of random processes is of significance in understanding their extreme values. I will illustrate this by several examples including cover times for random walks, maxima for two-dimensional discrete Gaussian free fields, and stochastic distance models. Our main results include (1) An approximation of the cover time on any graph up to a multiplicative constant by the maximum of the Gaussian free field, which yields a deterministic polynomial-time approximation algorithm for the cover time (D.-Lee-Peres 2010); the asymptotics for the cover time on a bounded-degree graph by the maximum of the GFF (D. 2011); a bound on the cover time fluctuations on the 2D lattice (D. 2011). (2) Exponential and doubly exponential tails for the maximum of the 2D GFF (D. 2011); some results on the extreme process of the 2D GFF (D.-Zeitouni, in preparation). (3) Critical and near-critical behavior for the mean-field stochastic distance model (D. 2011).
Title: Imaginary Geometry and the Gaussian Free Field
Abstract: The Schramm-Loewner evolution (SLE) is the canonical model of a non-crossing conformally invariant random curve, introduced by Oded Schramm in 1999 as a candidate for the scaling limit of loop erased random walk and the interfaces in critical percolation. The development of SLE has been one of the most exciting areas in probability theory over the last decade because Schramm's curves have now been shown to arise as the scaling limit of the interfaces of a number of different discrete models from statistical physics. In this talk, I will describe how SLE curves can be realized as the flow lines of a random vector field generated by the Gaussian free field, the two-time-dimensional analog of Brownian motion. I will also explain how this perspective can be used to prove several new results regarding the sample path behavior of SLE, in particular reversibility for kappa in (4,8). Based on joint works with Scott Sheffied.
Title: Directed random polymers and Macdonald processes
Abstract: The goal of the talk is to survey recent progress in understanding statistics of certain exactly solvable growth models, particle systems, directed polymers in one space dimension, and stochastic PDEs. A remarkable connection to representation theory and integrable systems is at the heart of Macdonald processes, which provide an overarching theory for this solvability. This is based off of joint work with Alexei Borodin.
Title: Path properties of the Schramm-Loewner Evolution.
Title: Extrema of branching Brownian motion
Abstract: Branching Brownian motion (BBM) on the real line is a particle system where particles perform Brownian motion and independently split into two independent Brownian particles after an exponential holding time. The statistics of extremal particles of BBM in the limit of large time are of interest for physicists and probabilists since BBM constitutes a borderline case, among Gaussian processes, where correlations affect the statistics. In this talk, I will start by reviewing results on the law of the maximum of BBM (the rightmost particle), and present new results on the joint distribution of particles close to the maximum. In particular, I will show how the approach can be used to prove ergodicity of the particle system. If time permits, I will explain how the program for BBM lays out a road map to understand extrema of log-correlated Gaussian fields such as the 2D Gaussian free field. This is joint work with A. Bovier and N. Kistler.
This is a special event. Billingsley Lectures on Probability in honor of Patrick Billingsley
Title: Large Deviations with Applications to Random Matrices and Random Graphs
Abstract: See it here.
Title: Large Deviations for an Unusual Sum
Abstract: See it here.
Title: Landscape of random functions in many dimensions via Random Matrix Theory.
Abstract: How many critical values a typical Morse function have on a high dimensional manifold? Could we say anything about the topology of its level sets? In this talk I will survey a joint work with Gerard Ben Arous and Jiri Cerny that addresses these questions in a particular but fundamental example. We investigate the landscape of a general Gaussian random smooth function on the N-dimensional sphere. These corresponds to Hamiltonians of well-known models of statistical physics, i.e spherical spin glasses. Using the classical Kac-Rice formula, this counting boils down to a problem in Random Matrix Theory. This allows us to show an interesting picture for the complexity of these random Hamiltonians, for the bottom of the energy landscape, and in particular a strong correlation between the index and the critical value. We also propose a new invariant for the possible transition between the so-called 1-step replica symmetry breaking and a Full Replica symmetry breaking scheme and show how the complexity function is related to the Parisi functional.
Title: Finite-rank deformations of Wigner matrices.
Abstract: The spectral statistics of large Wigner matrices are by now well-understood. They exhibit the striking phenomenon of universality: under very general assumptions on the matrix entries, the limiting spectral statistics coincide with those of a Gaussian matrix ensemble. I shall talk about Wigner matrices that have been perturbed by a finite-rank matrix. By Weyl's interlacing inequalities, this perturbation does not affect the large-scale statistics of the spectrum. However, it may affect eigenvalues near the spectral edge, causing them to break free from the bulk spectrum. In a series of seminal papers, Baik, Ben Arous, and Peche (2005) and Peche (2006) established a sharp phase transition in the statistics of the extremal eigenvalues of perturbed Gaussian matrices. At the BBP transition, an eigenvalue detaches itself from the bulk and becomes an outlier. I shall report on recent joint work with Jun Yin. We consider an NxN Wigner matrix H perturbed by an arbitrary deterministic finite-rank matrix A. We allow the eigenvalues of A to depend on N. Under optimal (up to factors of log N) conditions on the eigenvalues of A, we identify the limiting distribution of the outliers. We also prove that the remaining eigenvalues "stick" to eigenvalues of H, thus establishing the edge universality of H + A. On the other hand, our results show that the distribution of the outliers is not universal, but depends on the distribution of H and on the geometry of the eigenvectors of A. As the outliers approach the bulk spectrum, this dependence is washed out and the distribution of the outliers becomes universal.
Jonathan Mattingly - Duke University
Title: A Menagerie of Stochastic Stabilization
Abstract: A basic problem for a stochastic system is to show that it possesses a unique steady state which dictates the long term statistics of the system. Sometimes the existence of such a measure is the difficult part. One needs control of the excursions away from the systems typical scale. As in deterministic system, one popular method is the construction of a Lyapunov Function. In the stochastic setting there lack of systematic methods to construct a Lyapunov Function when the interplay between the deterministic dynamics and stochastic dynamics are important for stabilization. I will give some modest steps in this direction which apply to a number of cases. In particular I will show a system where an explosive deterministic system is stabilized by the addition of noise and examples of physical systems where it is not clear how the deterministic system absorbs the stochastic excitation with out blowing up.
Title: From random interlacements to coordinate and infinite cylinder percolation
Abstract: During the talk I will focus on the connectivity properties of three models with long (infinite) range dependencies: Random Interlacements, percolation of the vacant set in infinite rod model and Coordinate percolation. The latter model have polynomial decay in sub-critical and super-critical regime in dimension 3. I will explain the nature of this phenomenon and why it is difficult to handle these models technically. In the second half of the talk I will present key ideas of the multi-scale analysis which allows to reach some conclusions. At the end I will discuss applications and several open problems.
Title: Complete matchings and random matrix theory
Abstract: Over the last decade or so, it has been found that the distributions that first appeared in random matrix theory describe several objects in probability and combinatorics which do not come from matrix at all. We consider one such example from the so-called maximal crossing and nesting of random complete matchings of integers. We also discuss related non-intersecting process. This is a joint work with Bob Jenkins.
Title: A simplified proof of the relation between scaling exponents in first-passage percolation
Abstract: In first passage percolation, we place i.i.d. non-negative weights on the nearest-neighbor edges of Z^d and study the induced random metric. A long-standing conjecture gives a relation between two "scaling exponents": one describes the variance of the distance between two points and the other describes the transversal fluctuations of optimizing paths between the same points. This is sometimes referred to as the "KPZ relation." In a recent breakthrough work, Sourav Chatterjee proved this conjecture using a strong definition of the exponents. I will discuss work I just completed with Tuca Auffinger, in which we introduce a new and intuitive idea that replaces Chatterjee's main argument and gives an alternative proof of the relation. One advantage of our argument is that it does not require a certain non-trivial technical assumption of Chatterjee on the weight distribution.
Ofer Zeitouni - University of Minnessota
Title: Traveling waves, branching random walks, and the Gaussian free field
Abstract: I will discuss several aspects of Branching random walks and their relation with the KPP equation on the one hand, and the maximum of certain (two dimensional) Gaussian fields on the other. I will not assume any knowledge about either of these terms.
Title: The parafermionic observable in Schramm-Loewner Evolutions
Abstract: In recent years, work by Stanislav Smirnov and his co-authors has greatly advanced our understanding of discrete stochastic processes, such as self-avoiding walk and the Ising model, via the use of a tool known as the parafermionic observable. Much of that work has been done in order to show convergence of these models to Schramm-Loewner Evolutions (SLE) in the scaling limit, although very little work has been done on what the parafermionic observable is in SLE itself. In this talk I will introduce the parafermionic observable, and then discuss one possible generalization to the continuous setting. I will then briefly introduce SLE and compute its parafermionic observable, ending with a couple of open questions.
Title: The contact process on the complete graph with random, vertex-dependent infection rates.
Abstract: The contact process is an interacting particle system that is a very simple model for the spread of an infection or disease on a network. Traditionally, the contact process was studied on homogeneous graphs such as the integer lattice or regular trees. However, due to the non-homogeneous structure of many real-world networks, there is currently interest in studying interacting particle systems in non-homogeneous graphs and environments. In this talk, I consider the contact process on the complete graph, where the vertices are assigned (random) weights and the infection rate between two vertices is proportional to the product of their weights. This set-up allows for some interesting analysis of the process and detailed calculations of phase transitions and critical exponents.
Title: Universality for beta-ensembles.
Abstract: Wigner stated the general hypothesis that the distribution of eigenvalue spacings of large complicated quantum systems is universal in the sense that it depends only on the symmetry class of the physical system but not on other detailed structures. The simplest case for this hypothesis is for ensembles of large but finite dimensional matrices. Spectacular progress was done in the past decade to prove universality of random matrices presenting an orthogonal, unitary or symplectic invariance. These models correspond to log-gases with respective inverse temperature 1, 2 or 4. I will report on a joint work with L. Erd\"os and H.-T. Yau, which yields universality for the log-gases at arbitrary temperature. The involved techniques include a multiscale analysis and a local logarithmic Sobolev inequality.
Friday, Oct. 8, Fredrik Johansson Viklund, Columbia U.
Friday, Oct. 15, Midwest Probability Colloquium at Northwestern
Friday, Oct. 29, Tom Alberts, U. of Toronto,
Convergence of Loop-Erased Random Walk to SLE(2) in the
Natural Time Parameterization
I will discuss work in progress with Michael Kozdron and Robert Masson on the convergence of the two-dimensional loop-erased random walk process to SLE(2), with the time parameterization of the curves taken into account. This is a strengthening of the original Lawler, Schramm, and Werner result which was only for curves modulo a reparameterization. The ultimate goal is to show that the limiting curve is SLE(2) with the very specific natural time parameterization that was recently introduced in Lawler and Sheffield, and further studied in Lawler and Zhou. I will describe several possible choices for the parameterization of the discrete curve that should all give the natural time parameterization in the limit, but with the key difference being that some of these discrete time parameterizations are easier to analyze than the others.
Friday, Dec. 3, Pierre Nolin, Courant Institute
Connection probabilities and RSW-type bounds for the two-dimensional FK
Ising model
For two-dimensional independent percolation, Russo-Seymour-Welsh (RSW) bounds on crossing probabilities are an important a-priori indication of scale invariance, and they turned out to be a key tool to describe the phase transition: what happens at and near criticality. In this talk, we prove RSW-type uniform bounds on crossing probabilities for the FK Ising model at criticality, independent of the boundary conditions. A central tool in our proof is Smirnov's fermionic observable for the FK Ising model, that makes some harmonicity appear on the discrete level, providing precise estimates on boundary connection probabilities. We also prove several related results - including some new ones - among which the fact that there is no magnetization at criticality, tightness properties for the interfaces, and the value of the half-plane one-arm exponent. This is joint work with H. Duminil-Copin and C. Hongler.