## Probability and Statistical Physics Seminar

### The University of Chicago

The seminar is organized by Lucas Benigni, Ahmed Bou-Rabee, Ewain Gwynne, Gregory Lawler, and Xuan Wu. It takes place in person on Fridays at 2:30 pm in Eckhart 202, unless otherwise specified.

## Spring 2022 Seminars

• Friday, April 1: Milind Hegde - Columbia University

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.

• Friday, April 8: Philippe Sosoe - Cornell University

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.

• Friday, April 8: Johannes Alt - University of Geneva

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.

• Friday, April 15: Firas Rassoul-Agha - University of Utah

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.

• Friday, April 22: Sevak Mkrtchyan - University of Rochester

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.

• Friday, April 29: Lingfu Zhang - Princeton University

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.

• Thursday, May 5 4:30 Billingsley Lecture: Fredrik Viklund - KTH Royal Institute of Technology

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).

• Friday, May 6: Fredrik Viklund - KTH Royal Institute of Technology

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)

• Friday, May 13: Nathanaël Berestycki - University of Vienna

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.

• Friday, May 20: Vishesh Jain - Stanford University

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.

## Winter 2022 Seminars

• Friday, January 14: Mark Sellke - Stanford University

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 32 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.

• Friday, January 21: Promit Ghosal - Massachusetts Institute of Technology

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.

• Friday, January 28: Bob Hough - Stony Brook University

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.

• Friday, February 4: Tom Hutchcroft - California Institute of Technology

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.

• Friday, February 11: Minjae Park - Massachusetts Institute of Technology

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.

• Friday, February 18: Tom Kennedy - University of Arizona

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.

• Friday, February 25: Giorgio Cipolloni - Princeton University

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.

• Friday, March 4: Roger Van Peski - Massachusetts Institute of Technology

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.

• Friday, March 11: Sky Cao - Stanford University

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.

• Friday, March 18: TBA

Title: TBA

Abstract: TBA

## Fall 2021 Seminars

• Friday, October 01: Evgeni Dimitrov - Columbia University

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.

• Friday, October 08: Marianna Russkikh - MIT

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.

• Friday, October 15: Xin Sun - University of Pennsylvania

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.

• Friday, October 15: Jiaoyang Huang - Courant Institute, NYU

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.

• Friday, October 22: Morris Ang - MIT

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.

• Friday, October 29: Hao Shen - University of Wisconsin-Madison

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.

• Friday, November 05: Jacopo Borga - Stanford University

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.

• Friday, November 12: Paul Wiegmann - University of Chicago

Title: Geometric aspects of random matrix ensembles

Abstract: TBA

• Friday, November 19: Erik Bates - University of Wisconsin–Madison

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.

• Friday, December 03: Guillaume Remy - Columbia University

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.

• Friday, December 10: TBA

Title: TBA

Abstract: TBA

## Spring 2021 Seminars

• Friday, April 02: Nicholas A. Cook - Duke University

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.

• Friday, April 09: Chris Janjigian - Purdue University

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.

• Friday, April 16: Tatyana Shcherbina - University of Wisconsin-Madison

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.

• Friday, April 23: Maria Gordina - University of Connecticut

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.

• Friday, April 30: Hugo Falconet - Columbia University

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.

• Friday, May 07: Sourav Chatterjee - Stanford University

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.

• Friday, May 14: David Damanik - Rice University

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)

• Friday, May 21: Pierre Yves Gaudreau Lamarre - University of Chicago

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.

• Friday, May 28: Gérard Ben Arous - New York University

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.

## Winter 2021 Seminars

• Friday, January 15 Exceptional Time 9h-10h AM : Hubert Lacoin - IMPA

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 $X$ indexed by $ℝd$ whose covariance $K$ 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 $X$. This mathematical object is what is called complex Gaussian multiplicative chaos. The field $X$ not being a function, but a random Gaussian distribution, the expression $e\gamma Xdx$ 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 $e\gamma Xdx$ 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.

• Friday, January 22: Emma Bailey - University of Bristol

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.

• Friday, January 29: Reza Gheissari - University of California, Berkeley

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.

• Friday, February 05: Yi Sun - University of Chicago

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.

• Friday, February 12: Willem van Zuiljen - Weierstrass Institute

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.

• Friday, February 19: Juhan Aru - EPFL

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.

• Friday, February 26: Sourav Sarkar - University of Toronto

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.

• Friday, March 05: Amol Aggarwal - Columbia University

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.

• Friday, March 12: Antti Knowles - University of Geneva

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.

## Fall 2020 Seminars

• Friday, October 2: Xuan Wu - University of Chicago

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.

• Friday, October 9: No seminar due to the Midwest Probability Colloquium

• Friday, October 16 : Gordon Slade - University of British Columbia

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.

• Friday, October 23: Pratima Hebbar - Duke University

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.

• Friday, October 30: Shirshendu Ganguly - University of California, Berkeley

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).

• Friday, November 6: Linan Chen - McGill University

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.

• Friday, November 13: Nina Holden - ETH Zürich

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.

• Friday, November 20: Jack Hanson - City College of New York, CUNY

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.

• Friday, December 4: Yilin Wang - MIT

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).

## Winter 2020 Seminars

• Friday, January 24: Patrick Lopatto - Harvard

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.

• Friday, January 31 (two talks)

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.

• Friday, February 21: Dmitry Dolgopyat - University of Maryland

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.

• Friday, February 28: Yizhe Zhu - University of California, San Diego

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.

## Fall 2019 Seminars

• Friday, October 4: Vishesh Jain - MIT

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.

• Friday, October 11: No seminar, but Midwest Probability Colloquium at Northwestern

• Friday, October 18: Oanh Nguyen - Princeton

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.

• Friday, October 25: Alisa Knizel - Columbia

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)

• Friday, November 1: Amir Dembo - Stanford

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.

• Friday, November 8: Shuwen Lou - Loyola University Chicago

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.

• Friday, November 15: no seminar

• Friday, November 22: Lucas Benigni - University of Chicago

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.

• Friday, November 29: no seminar (Thanksgiving)

• Friday, December 6: Ewain Gwynne - Cambridge

## Fall 2013 Seminars

• Friday, Oct 4th: Ofer Zeitouni - Courant Institute and Weizmann Institute of Science.

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.

• Friday, Oct 11th: Thirty-fifth Midwest Probability Colloquium

• Friday, Oct 18th: Amir Dembo - Stanford University.

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.

• Friday, Oct 25th: Erik Lundberg - Purdue University This talk was reschedule to Dec. 13th!

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.

• Friday, Nov 1st: Yashodhan Kanoria - Columbia Business School

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 beneÃ¯Â¬Ât 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.

• Friday, Nov 8th: Cris Moore - Santa Fe Institute.

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.

• Friday, Nov 15th: Shannon Starr - University of Alabama at Birmingham.

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.

• Friday, Nov 22nd: Roman Vershynin - University of Michigan.

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).

• Friday, Nov 29th: Thanksgiving

• Friday, Dec 6th: Shirshendu Chatterjee - Courant Institute

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.

• Friday, Dec 13th: Erik Lundberg - Purdue University

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.

## Spring 2013 Seminars

• Friday, Apr 5th: Alex Fribergh - Universite de Toulouse.

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.

• Friday, Apr 12th: Yuval Peres - Microsoft Research

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).

• Friday, Apr 12th (4:30-5:00): Yuval Peres - Microsoft Research

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.

• Wednesday, Apr 24th 4pm - 5pm at the CAMP seminar: Grigorios Pavliotis - Imperial College London.

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.

• Thursday, May 2nd: Persi Diaconis - Stanford University

This is a special event. Billingsley Lectures on Probability in honor of Professor Billingsley.

• Friday, May 3rd: Persi Diaconis - Stanford University

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.

• Friday, May 10th: Tim Austin - New York University

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.

• Friday, May 17th: Nike Sun - Stanford University

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.

• Friday, May 24th: Lionel Levine - Cornell University

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!).

• Friday, May 31st: Jonathan Weare - University of Chicago

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.

• Friday, Jun 14th: Firas Rassoul-Agha - University of Utah

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.

• Friday, July 19th: Louigi Addario-Berry - McGill University.

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.

## Winter 2013 Seminars

• Friday, Feb 1st (1:30pm to 2:30pm): Marek Biskup - UCLA

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).

• Friday, Feb 1st (2:30pm to 3:30pm): Fredrik Viklund - Columbia University

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.

• Friday, Feb 8th: James Lee - University of Washington

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.

• Friday, Feb 15th: Michelle Castellana - Princeton University

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.

• Friday, Feb 22nd: Jack Hanson - Princeton University

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.

• Friday, Mar 1st: Vadim Gorin - M.I.T.

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.

• Friday, Mar 8th: No seminar.

• Friday, Mar 15th: Alice Guionnet - M.I.T.

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.

## Fall 2012 Seminars

• Friday, Oct 5th: Wei-Kuo Chen - University of Chicago

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.

• Friday, Oct 12th: Thirty-fourth Midwest Probability Colloquium

• Friday, Oct 19th: Gerard Ben Arous - Courant Institute

Abstract: This seminar was canceled. It will be rescheduled.

• Friday, Oct 26th: Allan Sly - UC Berkeley

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.

• Friday, Dec 7th: Brian Rider - University of Colorado Boulder

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.)

• Friday, Nov 2nd: Gregorio Moreno Flores - University of Wisconsin

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).

• Friday, Nov 9th: Milton Jara - IMPA

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 GonÃƒÂ§alves.

• Friday, Nov 16th: Mohammad Abbas Rezaei - University of Chicago

Title: SLE curves and natural parametrization

• Friday, Nov 23rd: Thanksgiving.

• Friday, Nov 30th: Joe Neeman - UC Berkeley

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.

• Friday, Dec 7th: Brian Rider - University of Colorado Boulder

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.)

## Winter/Spring 2012 Seminars

• Friday, Jan 20: Jian Ding - Stanford University

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).

• Friday, Feb 10: Jason Miller - Microsoft Research -Redmond

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.

• Friday, Mar 9: Ivan Corwin - Microsoft Research - MIT

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.

• Friday, April 13th: Brent Werness - University of Chicago

Title: Path properties of the Schramm-Loewner Evolution.

• Friday, May 11: L.P. Arguin - Univesite de Montreal

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.

• Thursday, May 31: S.R. Srinivasa Varadhan - Courant Institute of Mathematical Sciences at New York University

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.

• Friday, June 1st: S.R. Srinivasa Varadhan - Courant Institute of Mathematical Sciences at New York University

Title: Large Deviations for an Unusual Sum

Abstract: See it here.

## Fall 2011 Seminars

• Friday, Sep 30: Antonio Auffinger - University of Chicago

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.

• Friday, Oct 7: Antti Knowles - Harvard University

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.

• Friday, Oct. 14, Midwest Probability Colloquium at Northwestern

• Tuesday, Oct 18: Scientific and Statistical Computing Seminar (3:00 in Eckhart 207)

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.

• Friday, Oct 21: Vladas Sidoravicius - IMPA

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.

• Friday, Nov 4: Jinho Baik - University of Michigan

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.

• Friday, Nov 11: Michael Damron - Princeton University

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.

• Wednesday, Nov 16: CAMP/ Nonlinear PDEs Seminar (4pm in Eckhart 202)

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.

• Friday, Nov 18: Brent Werness - University of Chicago

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.

• Friday, Nov 25: Thanksgiving holiday. No seminar.

• Friday, Dec 2: Jonathon Peterson - Purdue University 1:30 pm!!

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.

• Friday, Dec 9: Paul Bougarde - Harvard University

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.

### Past Seminars

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.