## Probability and Statistical Physics Seminar

### University of Chicago

The probability seminar is organized by Vivian Healey, Steven Lalley, Gregory Lawler and Xinyi Li. It takes place on Fridays at 2:30 pm in Eckhart 202, unless otherwise specified.

## Winter 2019 Seminars

• Monday, Jan 14, 1:30pm, Eckhart 207 (special day, time and room!): Huy V. Tran - TU Berlin

Title: A support theorem for SLE curves

Abstract: SLE curves are an important family of random curves in the plane. They share many similarities with solutions of SDE (in particular, with Brownian motion). Any question asked for the latter can be asked for the former. Inspired by that, Yizheng and I investigate the support for SLE curves. In this talk, I will explain our theorem with more motivation and idea.

• Friday, Jan 25: Ilia Binder - University of Toronto

Title: Large concentration of harmonic measure: discrete and continuous case.

Abstract: I will discuss the sharp bounds on the multifractal spectrum of planar harmonic measure and their continuous analogues. In particular, I will talk about the refinements of Beurling estimates on the concentration of harmonic measure.

• Friday, Feb 1: Jiaoyang Huang - Harvard

Title: Invertibility of adjacency matrices for random d-regular graphs

Abstract: The singularity problem of random matrices asks the probability that a given discrete random matrix is singular. The first such result was obtained by Komlós in 1967. He showed a Bernoulli random matrix is singular with probability o(1). This question can be reformulated for the adjacency matrices of random graphs, either directed or undirected. The most challenging case is when the random graph is sparse. In this talk, I will prove that for random directed and undirected d-regular graphs, their adjacency matrices are invertible with high probability for all d>=3. The idea is to study the adjacency matrices over a finite field, and the proof combines a local central limit theorem and a large deviation estimate.

• Friday, Feb 8: Balint Virag - University of Toronto

Title: The directed landscape

Abstract: The longest increasing subsequence in a random permutation, the second class particle in TASEP, and semi-discrete polymers at zero temperature have the same scaling limit: a random function with Holder exponent 2/3-. This limit can be described in terms of the directed landscape, a random metric at the heart of the Kardar-Parisi-Zhang universality class. Joint work with Duncan Dauvergne and Janosch Ortmann.

• Friday, Feb 15: Mateo Wirth - University of Pennsylvania

Title: Chemical distance for level-set percolation of planar metric-graph Gaussian free field

Abstract: We consider percolation of the level-sets of a metric-graph Gaussian free field on a box of the planar integer lattice. We provide an upper bound for the length of the shortest path joining the boundary components of a macroscopic annulus. The bound holds with high probability conditioned on connectivity and is sharp up to a poly-logarithmic factor with exponent one-quarter. This is joint work with Jian Ding.

• Friday, Feb 22: Elton Hsu - Northwestern

Title: Bismut's Interpolation Between Riemannian Brownian Motion and Geodesic Flow

Abstract: A few years ago Bismut found a natural family of diffusion processes that interpolates between Brownian motion and the geodesic flow on a compact Riemannian manifold. The convergence to Brownian motion (together with an attendant Gaussian field) at one end of the parameter interval is nontrivial and proved in the weak sense of finite dimensional marginal distributions. By slightly extending the traditional setting for stochastic calculus on manifold we show that the convergence can be realized as a strong one in the path space. Such a more precise convergence to Brownian motion will help us in understanding asymptotic behavior of some classical functional inequalities for path spaces.

• Friday, March 1: Lionel Levine - Cornell

Title: Random walks with local memory

Abstract: The theme of this talk is walks in a random environment of signposts'' altered by the walker. I'll focus on three related examples:

1. Rotor walk on Z^2. Your initial signposts are independent with the uniform distribution on {North,East,South,West}. At each step you rotate the signpost at your current location clockwise 90 degrees and then follow it to a nearest neighbor. Priezzhev et al. conjectured that in n such steps you will visit order n^{2/3} distinct sites. I'll outline an elementary proof of a lower bound of this order. The upper bound, which is still open, is related to a famous question about the path of a light ray in a grid of randomly oriented mirrors. This part is joint work with Laura Florescu and Yuval Peres.

2. p-rotor walk on Z. In this walk you flip the signpost at your current location with probability 1-p and then follow it. I'll explain why your scaling limit will be a Brownian motion perturbed at its extrema. This part is joint work with Wilfried Huss and Ecaterina Sava-Huss.

3. p-rotor walk on Z^2. Rotate the signpost at your current location clockwise with probability p and counterclockwise with probability 1-p, and then follow it. This walk organizes'' its environment by destroying cycles of signposts. A native environment -- stationary in time, from your perspective as the walker -- is an orientation of the uniform spanning forest, plus one additional edge. This part is joint work with Swee Hong Chan, Lila Greco, and Peter Li.

• Friday, March 8: Qiang Zeng - CUNY Queens College

Title: 2-RSB for spherical mixed p-spin models at zero temperature

Abstract: Historically, the method of replica symmetry breaking (RSB) was introduced by Parisi to study mean field spin glass models. This method has played indispensable role in the physical deduction of Parisi formula. Mathematically, the level of RSB corresponds to the number of points in the support of the Parisi measure. There have been known Replica Symmetric, 1-step RSB and Full-step RSB 'natural' spin glass models. However, k-step RSB for finite k>1 is less well understood. In this talk, I will show that certain spherical mixed p-spin glass models are 2-step RSB at zero temperature and present consequences in the energy landscape. This talk is based on joint work with Antonio Auffinger.

## Spring 2019 Seminars

• Friday, Apr 5: Edwin Perkins - UBC

Title: On the range of lattice models in high dimensions.

Abstract: We investigate the scaling limit of the {\em range} (the set of visited vertices) for a general class of critical lattice models, starting from a single initial particle at the origin. Conditions are given on the random sets and an associated ancestral relation" under which, conditional on long term survival, the rescaled ranges converge weakly to the range of super-Brownian motion as random sets. These hypotheses also give precise asymptotics for the limiting behaviour of the probability of exiting a large ball. Applications include voter models, contact processes, oriented percolation and lattice trees. This is joint work with Mark Holmes and also features work of Akira Sakai and Gordon Slade.

• Friday, Apr 12: Jonathan Novak - UCSD

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• Friday, Apr 19: Atilla Yilmaz - Temple

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• Friday, Apr 26: Joe Chen - Colgate University

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• Friday, May 3: Charles M. Newman - NYU

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• Friday, May 17: Joan Lind - University of Tennessee, Knoxville

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• Friday, May 24: Yu Zhang - Colorado

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• Friday, June 7: Jun Yin - UCLA

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## Fall 2018 Seminars

• Friday, Aug 31: Daisuke Shiraishi - Kyoto University.

Title: Natural parametrization for the scaling limit of loop-erased random walk in three dimensions

Abstract: We will consider loop-erased random walk (LERW) and its scaling limit in three dimensions. Gady Kozma (2007) shows that as the lattice spacing becomes finer, LERW in three dimensions converges weakly to a random compact set with respect to the Hausdorff distance. We will show that 3D LERW parametrized by renormalized length converges in the lattice size scaling limit to Kozma's scaling limit parametrized by some suitable measure on it with respect to the uniform norm. This is based on joint works with Xinyi Li (University of Chicago).

• Friday, Oct 5: Robin Pemantle - University of Pennsylvania

Title: Percolation on random trees (joint work with Marcus Michelen and Josh Rosenberg)

Abstract: Let T be a tree chosen from Galton-Watson measure and let {U_v} be IID uniform [0,1] random variables associated with the edges between each vertex and its parent. These define coupled Bernoulli percolation processes, as well as an invasion percolation process. We study quenched properties of these percolations: properties conditional on T that hold for almost every T. The invasion cluster has a backbone decomposition which is Markovian if you put on the right blinders. Under suitable moment conditions, the law of the (a.s. unique) backbone ray is absolutely continuous with respect to limit uniform measure. The quenched survival probabilities are smooth in the supercritical region p > p_c. Their behavior as p -> p_c depends on moment assumptions for the offspring distribution.

• Friday, Oct 12: Jonathon Peterson - Purdue University

Title: Quantitative CLTs for random walks in random environments

Abstract: The Berry-Esseen estimates give quantitative error estimates on the CLT for sums of i.i.d. random variables, and the polynomial decay rate for the error depends on moment bounds of the i.i.d. random variables with the optimal $1/\sqrt{n}$ rate of convergence obtained under a third moment assumption. In this talk we will prove quantitative error bounds for CLTs of random walks in random environments (RWRE). That is, for certain models of RWRE it is known that the position of the random walk has a Gaussian limiting distribution, and we obtain quantitative error estimates on the rate of convergence to the Gaussian distribution for such RWRE. This talk is based on joint works with Sungwon Ahn and Xiaoqin Guo.

• October 18-20: (no seminar, but) Midwest Probability Colloquium @Northwestern University.

• Friday, Oct 26: Will Perkins - University of Illinois at Chicago

Title: Algorithmic Pirogov-Sinai theory

Abstract: We develop efficient algorithms to approximate the partition function and sample from the hard-core and Potts models on lattices at sufficiently low temperatures in the phase coexistence regime. In contrast, the Glauber dynamics are known to take exponential time to mix in this regime. Our algorithms are based on the cluster expansion and Pirogov-Sinai theory, classical tools from statistical physics for understanding phase transitions, as well as Barvinok's approach to polynomial approximation. Joint work with Tyler Helmuth and Guus Regts.

• Friday, Nov 2: Jason Schweinsberg - UCSD

Title: Yaglom-type limit theorems for branching Brownian motion with absorption

Abstract: We consider one-dimensional branching Brownian motion in which particles are absorbed at the origin. We assume that when a particle branches, the offspring distribution is supercritical, but the particles are given a critical drift towards the origin so that the process eventually goes extinct with probability one. We establish precise asymptotics for the probability that the process survives for a large time t, improving upon a result of Kesten (1978) and Berestycki, Berestycki, and Schweinsberg (2014). We also prove a Yaglom-type limit theorem for the behavior of the process conditioned to survive for an unusually long time, which also improves upon results of Kesten (1978). An important tool in the proofs of these results is the convergence of branching Brownian motion with absorption to a continuous state branching process.

• Thursday, Nov 8: Billingsley lecture by Allan Sly - Princeton University

Title: Phase transitions of random constraint satisfaction problems

Abstract: Random constraint satisfaction problems encode many interesting questions in the study of random graphs such as the chromatic and independence numbers. Ideas from statistical physics provide a detailed description of phase transitions and properties of these models. We will discuss the one step replica symmetry breaking transition that many such models undergo.

• Friday, Nov 9: Allan Sly - Princeton University

Title: Coalescence of polymers in Last Passage Percolation

Abstract: We will discuss new bounds on the time for polymers in first passage percolation to coalesce. As a consequence we will prove that there are no non-trivial bi-geodesics. The methods will combine bounds given by exactly solvable calculations with tools from percolation theory.

• Friday, Nov 16: Martin Barlow - University of British Columbia

Title: Stability of the elliptic Harnack Inequality

Abstract: Following the work of Moser, as well as de Giorgi and Nash, Harnack inequalities have proved to be a powerful tool in PDE as well as in probability. In the early 1990s Grigor'yan and Saloff-Coste gave a characterisation of the parabolic Harnack inequality (PHI). This characterisation implies that the PHI is stable under bounded perturbation of weights, as well as rough isometries. In this talk we prove the stability of the EHI. The proof uses the concept of a quasi symmetric transformation of a metric space, and the introduction of these ideas to Markov processes suggests a number of new problems. This is joint work with Mathav Murugan (UBC).

• Friday, Nov 30: Renming Song - University of Illinois Urbana-Champaign

Title: Factorizations and estimates of Dirichlet heat kernels for non-local operators with critical killings

Abstract: In this talk I will discuss heat kernel estimates for critical perturbations of non-local operators. To be more precise, let $X$ be the reflected $\alpha$-stable process in the closure of a smooth open set $D$, and $X^D$ the process killed upon exiting $D$. We consider potentials of the form $\kappa(x)=C\delta_D(x)^{-\alpha}$ with positive $C$ and the corresponding Feynman-Kac semigroups. Such potentials do not belong to the Kato class. We obtain sharp two-sided estimates for the heat kernel of the perturbed semigroups. The interior estimates of the heat kernels have the usual $\alpha$-stable form, while the boundary decay is of the form $\delta_D(x)^p$ with non-negative $p\in [\alpha-1, \alpha)$ depending on the precise value of the constant $C$. Our result recovers the heat kernel estimates of both the censored and the killed stable process in $D$. Analogous estimates are obtained for the heat kernel of the Feynman-Kac semigroup of the $\alpha$-stable process in ${\mathbf R}^d\setminus \{0\}$ through the potential $C|x|^{-\alpha}$. All estimates are derived from a more general result described as follows: Let $X$ be a Hunt process on a locally compact separable metric space in a strong duality with $\widehat{X}$. Assume that transition densities of $X$ and $\widehat{X}$ are comparable to the function $\widetilde{q}(t,x,y)$ defined in terms of the volume of balls and a certain scaling function. For an open set $D$ consider the killed process $X^D$, and a critical smooth measure on $D$ with the corresponding positive additive functional $(A_t)$. We show that the heat kernel of the the Feynman-Kac semigroup of $X^D$ through the multiplicative functional $\exp(-A_t)$ admits the factorization of the form ${\mathbf P}_x(\zeta >t)\widehat{\mathbf P}_y(\widehat{\zeta}>t)\widetilde{q}(t,x,y)$. This is joint work with Soobin Cho, Panki Kim and Zoran Vondracek.

• Friday, Nov 30: Yuri Bakhtin - NYU

Title: Ergodic theory of the stochastic Burgers equation

Abstract: The stochastic Burgers equation is one of the basic evolutionary SPDEs related to fluid dynamics and KPZ, among other things. The ergodic properties of the system in the compact space case were understood in 2000's. With my coauthors, Eric Cator, Kostya Khanin, Liying Li, I have been studying the noncompact case. The one force - one solution principle has been proved for positive and zero viscosity. The analysis is based on long-term properties of action minimizers and polymer measures. The latest addition to the program is the convergence of infinite volume polymer measures to Lagrangian one-sided minimizers in the limit of vanishing viscosity (or, temperature) which results in the convergence of the associated global solutions and invariant measures.

• Friday, Dec 7, 3:00 PM: Cheng Ouyang - University of Illinois at Chicago

Title: Local density estimate for a hypoelliptic SDE

Abstract: In a series of three papers in the 80’s, Kusuoka and Stroock developed a probabilistic program in order to obtain sharp bounds for the density function of a hypoelliptic SDE driven by a Brownian motion. We aim to investigate how their method can be used to study rough SDEs driven by fractional Brownian motions. In this talk, I will outline Kusuoka and Stroock’s approach and point out where the difficulties are in our current setting. The talk is based on an ongoing project with Xi Geng and Samy Tindel.

## Spring 2018 Seminars

• Friday, Mar 30: Alan Hammond - University of California at Berkeley

Title: The weight, geometry and coalescence of scaled polymers in Brownian last passage percolation

Abstract: In last passage percolation (LPP) models, a random environment in the two-dimensional integer lattice consisting of independent and identically distributed weights is considered. The weight of an upright path is said to be the sum of the weights encountered along the path. A principal object of study are the polymers, which are the upright paths whose weight is maximal given the two endpoints. Polymers move in straight lines over long distances with a two-thirds exponent dictating fluctuation. It is natural to seek to study collective polymer behaviour in scaled coordinates that take account of this linear behaviour and the two-third exponent-determined fluctuation. We study Brownian LPP, a model whose integrable properties find an attractive probabilistic expression. Building on a study arXiv:1609.02971 concerning the decay in probability for the existence of several near polymers with common endpoints, we demonstrate that the probability that there exist k disjoint polymers across a unit box in scaled coordinates has a superpolynomial decay rate in k. This result has implications for the Brownian regularity of the scaled polymer weight profile begun from rather general initial data.

• Friday, Apr 13: Yuan Zhang - Texas A & M University

Title: Stationary Harmonic Measure and DLA in the Upper Half Plane

Abstract: In this talk, we introduce the stationary harmonic measure in the upper half plane. By bounding this measure, we are able to define both the discrete and continuous time diffusion limit aggregation (DLA) in the upper half plane with absorbing boundary conditions. We prove that for the continuous model the growth rate is bounded from above by $o(2+\epsilon)$. When time is discrete, we also prove a better upper bound of $o(2/3+\epsilon)$, on the maximum height of the aggregate at time $n$.

• Friday, Apr 20: Christopher Hoffman - University of Washington

Title: Geodesics in First-Passage Percolation

Abstract: First-passage percolation is a classical random growth model which comes from statistical physics. We will discuss recent results about the relationship between the limiting shape in first passage percolation and the structure of the infinite geodesics. This includes a solution to the midpoint problem of Benjamini, Kalai and Schramm. This is joint work with Daniel Ahlberg.

• Friday, April 27: Timo Seppalainen - University of Wisconsin-Madison

Title: Shifted weights and restricted path length in first-passage percolation

Abstract: First-passage percolation has remained a challenging field of study since its introduction in 1965 by Hammersley and Welsh. There are many outstanding open problems. Among these are properties of the limit shape and the Euclidean length of geodesics. This talk describes a convex duality between a shift of the edge weights and the length of the geodesic, together with related results on the regularity of the limit shape as a function of the shift. The talk is based on joint work with Arjun Krishnan (Rochester) and Firas Rassoul-Agha (Utah).

• Friday, May 4: Laura Eslava - Georgia Tech

Title: The giant component in a degree-bounded process

Abstract: Graph processes $(G(i),i\ge 0)$ are usually defined as follows. Starting from the empty graph on $n$ vertices, at each step $i$ a random edge is added from a set of available edges. For the $d$-process, edges are chosen uniformly at random among all edges joining vertices of current degree at most $d-1$. The fact that, during the process, vertices become 'inactive' when reaching degree $d$ makes the process depend heavily on its history. However, it shares several qualitative properties with the classical Erdos-Renyi graph process. For example, there exists a critical time $t_c$ at which a giant component emerges, whp (that is, the largest component in $G(tn)$ goes from logarithmic to linear order). In this talk we consider $d\ge 3$ fixed and describe the growth of the size of the giant component. In particular, we show that whp the largest component in $G((t_c+\eps)n)$ has asymptotic size $cn$, where $c\sim c_d \eps$ is a function of time $\eps$ as $\eps \to 0+$. The growth, linear in $\eps$, is a new common qualitative feature shared with the Erdos-Renyi graph process and can be generalized to hypergraph processes with different max-allowed degree sequences. This is work in progress jointly with Lutz Warnke.

• Friday, May 18: Zhongyang Li - University of Connecticut

Title: Phase transitions in the 1-2 model

Abstract: A configuration in the 1-2 model is a subgraph of the hexagonal lattice, in which each vertex is incident to 1 or 2 edges. By assigning weights to configurations at each vertex, we can define a family of probability measures on the space of these configurations, such that the probability of a configuration is proportional to the product of weights of configurations at vertices. We study the phase transition of the model by investigating the probability measures with varying weights. We explicitly identify the critical weights, in the sense that the edge-edge correlation decays to 0 exponentially in the subcritical case, and converges to a non-zero constant in the supercritical case, under the limit measure obtained from torus approximation. These results are obtained by a novel measure-preserving correspondence between configurations in the 1-2 model and perfect matchings on a decorated graph, which appears to be a more efficient way to solve the model, compared to the holographic algorithm used by computer scientists to study the model. The major difficulty here is the absence of stochastic monotonicity.

• Friday, May 25: Jinho Baik - University of Michigan

Title: Fluctuations of the free energy of spherical Sherrington-Kirkpatrick model

Abstract: Consider the question of finding the maximum of a random polynomial defined on a closed manifold or a finite graph. Spherical Sherrington-Kirkpatrick (SSK) model is a finite temperature version of this question when the underlying space is a sphere. The free energy is the finite temperature version of the maximum value. The limit of the free energy as the dimension of the sphere becomes infinity is known by the works of Parisi, Cristanti, Sommers, and Talagrand. In this talk we consider the fluctuations when the polynomial is a symmetric quadratic function. We use a connection to random matrices and obtain limit theorems. This is a joint work with Ji Oon Li and Hao Wu.

• Wednesday, May 30 ( 3pm -4pm ) : Colloquium by Greg Lawler - The University of Chicago.

Title: Conformally Invariant Paths and Loops

Abstract: Abstract: There has been incredible progress in the last twenty years in the rigorous understanding of two-dimensional critical systems in statistical physics. I will give an overview with an emphasis on several related models: loop-erased random walk, spanning trees and corresponding loop soup; fractal paths and loops arising in critical systems (Schramm-Loewner evolution); the Gaussian free field and functions thereof (quantum gravity). I will also discuss some challenges for the future. This talk is intended for a general audience - it is not assumed that the audience is familiar with these terms.

• Friday, June 1: Alexander Drewitz - University of Cologne

Title: Phase transitions in some percolation models with long-range correlations on general graphs

Abstract: Abstract: We consider two fundamental percolation models with long-range correlations on reasonably general and well-behaved transient graphs: The Gaussian free field and (the vacant set) of Random Interlacements. Both models have been the subject of intensive research during the last years and decades, on $\Z^d$ as well as on some more general graphs. We consider their percolation phase transition and investigate a couple of interesting properties of their critical parameters, in particular the existence of a phase transition. This talk is based on joint works with A. Prevost (Koeln) and P.-F. Rodriguez (Los Angeles).

## Winter 2018 Seminars

• [CANCELLED, RESCHEDULED TO May 18] Friday, Jan 5: Zhongyang Li - University of Connecticut

• Friday, Jan 12: Shirshendu Ganguly - UC Berkeley NB: this seminar takes place exceptionally at Eckhart 133!!

Title: Understanding rare events in models of statistical mechanics

Abstract: Statistical mechanics models are ubiquitous at the interface of probability theory, information theory, and inference problems in high dimensions. In this talk we will focus on sparse graphs, and polymers on lattices; two canonical models in natural sciences. The study of large deviations is intimately related to the understanding of such models. We will consider the rare events that a sparse random network has an atypical number of certain local structures and that a polymer in random media has atypical weight. Conditioning on such events can produce different, ranging from local to more global, geometric effects. We will discuss some such results obtained, relying on a variety of entropy theoretic, combinatorial, and analytic tools.

• [CANCELLED-RESCHEDULED TO April 13] Friday, Jan 26: Yuan Zhang - Texas A & M University

Title: Stationary Harmonic Measure and DLA in the Upper Half Plane

Abstract: In this talk, we introduce the stationary harmonic measure in the upper half plane. By bounding this measure, we are able to define both the discrete and continuous time diffusion limit aggregation (DLA) in the upper half plane with absorbing boundary conditions. We prove that for the continuous model the growth rate is bounded from above by $o(2+\epsilon)$. When time is discrete, we also prove a better upper bound of $o(2/3+\epsilon)$, on the maximum height of the aggregate at time $n$.

• Friday, Feb 2: Samuel Watson - Brown University

Title: Relating a classical planar map embedding algorithm to Liouville quantum gravity and SLE(16)

Abstract: In 1990, Walter Schnyder introduced a class of 3-spanning-tree decompositions of a simple triangulation to describe a combinatorially natural grid embedding algorithm for planar maps. It turns out that a uniformly sampled Schnyder-wood-decorated triangulation on n vertices converges as n tends to infinity to a random fractal surface, called a Liouville quantum gravity (LQG), together with a triple of intertwined fractal curves known as SLE(16). We will motivate this result by describing Schnyder’s algorithm and discussing some history of random planar map convergence results, and we will also introduce LQG and SLE and explain their role in the story.

• Friday, Feb 16: Omer Angel - University of British Columbia

Title: New perspectives on Mallows permutations

Abstract: I will discuss two projects concerning Mallows permutations, with Ander Holroyd, Tom Hutchcroft and Avi Levy. First, we relate the Mallows permutation to stable matchings, and percolation on bipartite graphs. Second, we study the scaling limit of the cycles in the Mallows permutation, and relate it to diffusions and continuous trees.

• Friday, Feb 23 Julian Gold - Northwestern University

Title: Dynamical freezing in a spin glass system with logarithmic correlations

Abstract: We consider a continuous time random walk on the two-dimensional discrete torus, whose motion is governed by the discrete Gaussian free field on the corresponding box acting as a potential. More precisely, at any vertex the walk waits an exponentially distributed time with mean given by the exponential of the field and then jumps to one of its neighbors, chosen uniformly at random. We prove that throughout the low-temperature regime and at in-equilibrium timescales, the process admits a scaling limit as a spatial K-process driven by a random trapping landscape, which is explicitly related to the limiting extremal process of the field. Alternatively, the limiting process is a supercritical Liouville Brownian motion with respect to the continuum Gaussian free field on the box. Joint work with Aser Cortines (University of Zurich) and Oren Louidor (Technion).

• Friday, Mar 2: Alexander Fribergh - Universite de Montreal

Title: The ant in high dimensional labyrinths

Abstract: One of the most famous open problem in random walks in random environments is to understand the behaviour of a simple random walk on a critical percolation cluster, a model known as the ant in the labyrinth. We will present new results on the scaling limit for the simple random walk on the critical branching random walk in high dimension which converges, after scaling, to the Brownian motion on the integrated super-brownian motion. In the light of lace expansion, we believe that the limiting behaviour of this model should be universal for simple random walks on critical structures in high dimension. In particular, recent progress show that similar results hold for lattice trees.

## Fall 2017 Seminars

• Friday, September 29: Eviatar Procaccia - Texas A&M University

Title: Stationary aggregation processes

Abstract: In this talk I'll introduce stationary versions of known aggregation models e.g., DLA, Hastings Levitov, IDLA and Eden. Using the additional symmetry and ergodic theory, one obtains new geometric insight on the aggregation processes.

• Thursday Oct 5: Billingsley lecture! by Yuval Peres -Microsoft Research and UC Berkeley

Title: Gravitational allocation to uniform points on the sphere

Abstract: Given n uniform points on the surface of a two-dimensional sphere, how can we partition the sphere fairly among them? "Fairly" means that each region has the same area. It turns out that if the given points apply a two-dimensional gravity force to the rest of the sphere, then the basins of attraction for the resulting gradient flow yield such a partition-with exactly equal areas, no matter how the points are distributed. (See the cover of the AMS Notices at http://www.ams.org/publications/journals/notices/201705/rnoti-cvr1.pdf.) Our main result is that this partition minimizes, up to a bounded factor, the average distance between points in the same cell. I will also present an application to almost optimal matching of n uniform blue points to n uniform red points on the sphere, connecting to a classical result of Ajtai, Komlos and Tusnady (Combinatorica 1984). Joint work with Nina Holden and Alex Zhai.

• Friday Oct 6: Yuval Peres -Microsoft Research and UC Berkeley

Title: The strange geometry of high-dimensional random spanning forests.

Abstract: The uniform spanning forest (USF) in the lattice Z^d, first studied by Pemantle (Ann. Prob. 1991), is defined as a limit of uniform spanning trees in growing finite boxes. Although the USF is a limit of trees, it might not be connected- Indeed, Pemantle proved that the USF in Z^d is connected if and only if d<5. In later work with Benjamini, Kesten, and Schramm (Ann. Math 2004) we extended this result, and showed that the component structure of the USF undergoes a phase transition every 4 dimensions: For dimensions d between 5 and 8 there are infinitely many trees, but any two trees are adjacent; for d between 9 and 12 this fails, but for every two trees in the USF there is an intermediary tree, adjacent to each of the them. And this pattern continues, with the number of intermediary trees required increasing by 1 every 4 dimensions. In this talk, I will show that this is not the whole story, and for d>8 the USF geometry undergoes a qualitative change every time the dimension increases by 1. (Joint work with Tom Hutchcroft.)

• October 12-14: (no seminar, but) Midwest probability colloquium @Northwestern University.

• Friday, October 20: Paul Bourgade - NYU

Title: Random matrices, the Riemann zeta function and trees

Abstract: Fyodorov, Hiary & Keating have conjectured that the maximum of the characteristic polynomial of random unitary matrices behaves like extremes of log-correlated Gaussian fields. This allowed them to predict the typical size of local maxima of the Riemann zeta function along the critical axis. I will first explain the origins of this conjecture, and then outline the proof for the leading order of the maximum, for unitary matrices and the zeta function. This talk is based on joint works with Arguin, Belius, Radziwill and Soundararajan.

• Friday, November 10: Vadim Gorin - MIT and Russian Academy of Sciences

Title: Local limits of Random Sorting Networks

Abstract: A sorting network is a shortest path between 12..n and n..21 in the Cayley graph of the symmetric group spanned by swaps of adjacent letters. We will discuss the bulk local limit of the swap process of uniformly random sorting networks and encounter universal distributions of the random matrix theory, including the celebrated Gaudin-Mehta law, which describes the energy level spacings in heavy nuclei.

• [CANCELLED] Friday, Nov 17: Christopher Hoffman - University of Washington

Title: Geodesics in First-Passage Percolation

Abstract: First-passage percolation is a classical random growth model which comes from statistical physics. We will discuss recent results about the relationship between the limiting shape in first passage percolation and the structure of the infinite geodesics. This incudes a solution to the midpoint problem of Benjamini, Kalai and Schramm. This is joint work with Gerandy Brito and Daniel Ahlberg.

• Friday, Dec 1: Wenpin Tang - UCLA

Title: Regenerative permutations: Mallows(q) and Riemann zeta function

Abstract: In this talk we discuss regenerative permutations on integers, with emphasis on two particular models: p-shifted and P-biased permutations. When p is the geometric distribution, the p-shifted permutations appear to be the limit of Mallows permutation model. We generalize and simplify previous work of Gnedin and Olshanski. The P-biased permutations are reminiscent of successive sampling in Bayesian statistics. Interestingly, some zeta formulas appear in the evaluation of renewal quantities of GEM-biased permutations. This is based on joint work with Jean-Jil Duchamps and Jim Pitman.

## Spring 2017 Seminars

• Tuesday, March 28, 3pm - 4pm, Ryerson 251 (Combinatorics and Theoretical Computer Science Seminar): James Lee - University of Washington.

Title: Extremal metrics, eigenvalues, and graph separators

• Wednesday, March 29, 3pm - 4pm, Eckhart 206 (Colloquium).

Title: Discrete conformal metrics and spectral geometry on distributional limits

• Friday, March 31: Antonio Auffinger - Northwestern University.

Title: The SK model is FRSB at zero temperature

Abstract: In the early 80's, the physicist Giorgio Parisi wrote a series of ground breaking papers where he introduced the notion of replica symmetry breaking. His powerful insight allowed him to predict a solution for the SK model by breaking the symmetry of replicas infinitely many times. In this talk, we will prove Parisi's prediction at zero temperature for the mixed p-spin model, a generalization of the SK model. We will show that at zero temperature the functional order parameter is full-step replica symmetry breaking (FRSB). We will also describe the importance of this result for the description of the energy landscape. Based on recent works with Wei-Kuo Chen (U. of Minnesota) and Qiang Zeng (Northwestern U.).

• Friday, April 7: Joseph Yukich - Lehigh University.

Title: Limit theory for statistics of random geometric structures

Abstract: Questions arising in stochastic geometry and applied geometric probability are often understood in terms of the behavior of statistics of large random geometric structures.Such structures arise in diverse settings and include: (i) Point processes of dependent points in R^d, including determinantal, permanental, and Gibbsian point sets, as well as the zeros of Gaussian analytic functions, (ii) Simplicial complexes in topological data analysis, (iii) Graphs on random vertex sets in Euclidean space, (iv) Random polytopes generated by random data. Global features of geometric structures are often expressible as a sum of local contributions. In general the local contributions have short range spatial interactions but complicated long range dependence. In this survey talk we review stabilization'' methods for establishing the limit theory for statistics of geometric structures. Stabilization provides conditions under which the behavior of a sum of local contributions is similar to that of a sum of independent identically distributed random variables.

• Friday, April 14: Lisa Hartung - New York University.

Title: The Structure of Extreme Level Sets in Branching Brownian Motion

Abstract: We study the structure of extreme level sets of a standard one dimensional branching Brownian motion, namely the sets of particles whose height is within a fixed distance from the order of the global maximum. It is well known that such particles congregate at large times in clusters of order-one genealogical diameter around local maxima which form a Cox process in the limit. We add to these results by finding the asymptotic size of extreme level sets and the typical height and shape of those clusters which carry such level sets. We also find the right tail decay of the distribution of the distance between the two highest particles. These results confirm two conjectures of Brunet and Derrida (joint work with A. Cortines, O. Louidor).

• Friday, April 21: Qingsan Zhu - University of British Columbia

Title: Branching capacity and critical branching random walks

Abstract: In this talk, I will introduce branching capacity for any finite subset of Z^d (d>=5). It turns out to be an important subject in the study of critical branching random walks. I will discuss its connections with critical branching random walks from the following three perspectives: 1) the hitting probability of a set by critical branching random walk; 2) branching recurrence and branching transience; 3) the local limit of critical branching random walk in torus.

• Friday, April 28: Dapeng Zhan - Michigan State University

Title: SLE loop measures

Abstract: An SLE loop measure is a $\sigma$-finite measure on the space of loops, which locally looks like a Schramm-Loewner evolution (SLE) curve. In this work, we use Minkowski content (i.e., natural parametrization) of SLE to construct several types of SLE$_\kappa$ loop measures for $\kappa\in(0,8)$. First, we construct rooted SLE$_\kappa$ loop measures in the Riemann sphere $\widehat{\mathbb C}$, which satisfy M\"obius covariance, conformal Markov property, reversibility, and space-time homogeneity, when the loop is parameterized by its $(1+\frac \kappa 8)$-dimensional Minkowski content. Second, by integrating rooted SLE$_\kappa$ loop measures, we construct the unrooted SLE$_\kappa$ loop measure in $\widehat{\mathbb C}$, which satisfies M\"obius invariance and reversibility. Third, we extend the SLE$_\kappa$ loop measures from $\widehat{\mathbb C}$ to subdomains of $\widehat{\mathbb C}$ and to Riemann surfaces using Brownian loop measures, and obtain conformal invariance or covariance of these measures. Finally, using a similar approach, we construct SLE$_\kappa$ bubble measures in simply/multiply connected domains rooted at a boundary point. The SLE$_\kappa$ loop measures for $\kappa\in(0,4]$ give examples of Malliavin-Kontsevich-Suhov loop measures for all $c\le 1$. The space-time homogeneity of rooted SLE$_\kappa$ loop measures in $\widehat{\mathbb C}$ answers a question raised by Greg Lawler.

• Friday, May 19: Wei Wu - New York University

Title: Extremal and local statistics for gradient field models

Abstract: We study the gradient field models with uniformly convex potential (also known as the Ginzburg-Landau field) in two dimension. This is a log-correlated (but generally non-Gaussian) random field that arises in different branches of mathematical physics. Previous results (Naddaf-Spencer, and Miller) were focused on the CLT for the linear functionals of the field. In this talk I will describe more precise results on the marginal distribution and the extreme values of the field. Based on joint works with David Belius and Ron Peled.

• Friday, May 26: Rongfeng Sun - National University of Singapore

Title: Scaling limit of the directed polymer on Z^{2+1} in the critical window

Abstract: The directed polymer model on Z^{d+1} is the Gibbs transform of a directed random walk on Z^{d+1} in an i.i.d. random potential (disorder). It is known that the model undergoes a phase transition as the disorder strength varies, and disorder is relevant in d=1 and 2 in the sense that the presence of disorder, however weak, alters the qualitative behavior of the underlying random walk, with d=2 being the marginal case. For d=1, Alberts-Khanin-Quastel have shown that if the disorder strength tends to zero as a^{1/4} as the lattice spacing a tends to zero, then the partition functions converge to the solution of the Stochastic Heat Equation. We show that in the marginal dimension d=2, the partition functions admit non-trivial limits if the disorder strength scales as b/\sqrt{log 1/a}, with a transition at a critical point b_c. I will also discuss ongoing work in understanding the limit of the partition functions at b_c. Based on joint work with F. Caravenna and N. Zygouras.

• Friday, June 2: Vivian Healey - Brown University

Title: The Loewner Equation with Branching and the Continuum Random Tree

Abstract: In its most well-known form, the Loewner equation gives a correspondence between curves in the upper half-plane and continuous real functions (called the “driving function” for the equation). We consider the generalized Loewner equation, where the driving function has been replaced by a time-dependent real measure. In the first part of the talk, we investigate the delicate relationship between the driving measure and the generated hull, specifying a class of discrete random driving measures that generate hulls in the upper half-plane that are embeddings of trees. In the second part of the talk, we consider the scaling limit of these measures as the trees converge to the continuum random tree, with the goal of constructing an embedding of the CRT. We describe progress in this direction that has been obtained by analyzing the driving measures from an analytic standpoint, and we conclude by describing connections to the complex Burgers equation.

## Winter 2017 Seminars

• Friday, Feb 3: Tobias Johnson - New York University.

Title: Galton-Watson fixed points, tree automata, and interpretations

Abstract: Consider a set of trees such that a tree belongs to the set if and only if at least two of its root child subtrees do. One example is the set of trees that contain an infinite binary tree starting at the root. Another example is the empty set. Are there any other sets satisfying this property other than trivial modifications of these? I'll demonstrate that the answer is no, in the sense that any other such set of trees differs from one of these by a negligible set under a Galton-Watson measure on trees, resolving an open question of Joel Spencer's. This follows from a theorem that allows us to answer questions of this sort in general. All of this is part of a bigger project to understand the logic of Galton-Watson trees, which I'll tell you more about. Joint work with Moumanti Podder and Fiona Skerman.

• Friday, Feb 10: Daisuke Shiraishi - Kyoto University.

Title: On loops of Brownian motion

Abstract: We provide a decomposition of the trace of the Brownian motion into a simple path and an independent Brownian soup of loops that intersect the simple path. More precisely, we prove that any subsequential scaling limit of the loop erased random walk is a simple path (a new result in three dimensions), which can be taken as the simple path of the decomposition. In three dimensions, we also prove that the Hausdorff dimension of any such subsequential scaling limit lies in (1, 5/3]. We conjecture that our decomposition characterizes uniquely the law of the simple path. If so, our results would give a new strategy to the existence of the scaling limit of the loop erased random walk and its rotational invariance.

• Friday, Feb 17: Nina Holden - MIT.

Title: How round are the complementary components of planar Brownian motion?

Abstract:Consider a Brownian motion W in the complex plane started from 0 and run for time 1. Let A(1), A(2),... denote the bounded connected components of C-W([0,1]). Let R(i) (resp.\ r(i)) denote the out-radius (resp.\ in-radius) of A(i) for i \in N. Our main result is that E[\sum_i R(i)^2|\log R(i)|^\theta ]<\infty for any \theta<1. We also prove that \sum_i r(i)^2|\log r(i)|=\infty almost surely. These results have the interpretation that most of the components A(i) have a rather regular or round shape. Based on joint work with Serban Nacu, Yuval Peres, and Thomas S. Salisbury.

Friday, Mar 3: Wei Qian - ETH Zurich.

Title: Decomposition of Brownian loop-soup clusters

Abstract:We study the structure of Brownian loop-soup clusters in two dimensions. The first part of the talk is based on joint-work with Wendelin Werner. Among other things, we obtain the following decomposition of the clusters with critical intensity: When one conditions a loop-soup cluster by its outer boundary $l$ (which is known to be an SLE4-type loop), then the union of all excursions away from $l$ by all the Brownian loops in the loop-soup that touch $l$ is distributed exactly like the union of all excursions of a Poisson point process of Brownian excursions in the domain enclosed by $l$. In the second part of the talk, we condition a Brownian loop-soup cluster (of any intensity) on a portion $p$ of its boundary and show that the union of loops that touch $p$ satisfies the restriction property. This result implies that a phase transition occurs at c = 14/15 for the connectedness of the loops that touch $p$.

• Friday, Mar 10: Pierre-Francois Rodriguez - UCLA.

Title: Correlation inequalities for gradient fields and percolation

Abstract: We consider a class of massless gradient Gibbs measures, in dimension greater or equal to three, with uniformly convex potential (and non-convex perturbations thereof). A well-known example in this class is the Gaussian free field, which has received considerable attention in recent years. We derive a so-called decoupling inequality for these fields, which yields detailed information about their geometry, and the percolative and non-percolative phases of their level sets. An important aspect is the development of a suitable sprinkling technique, interesting in its own right, which we will discuss in some detail. Roughly speaking, it allows to dominate the strong correlations present in the model, and crucially relies on a particular representation of these correlations in terms of a random walk in a dynamic random environment, due to Helffer and Sjöstrand.

## Fall 2016 Seminars

• Thursday Oct 6: Billingsley lecture! @4:30pm, Eckhart 133 by Jean-Francois Le Gall -University Paris-Sud Orsay.

Title: Random planar geometry

• Friday Oct 7: Jean-Francois Le Gall -University Paris-Sud Orsay.

Title: First-passage percolation in random planar lattices

• October 13-15 (no seminar, but) Midwest probability colloquium @Northwestern University.

• Zygmund-Calderon Lectures by Martin Harier -University of Warwick.

Lecture 1: Taming infinities

Monday, October 24, 2016, 4:30pm–5:30pm, Ryerson 251

Abstract: Some physical and mathematical theories have the unfortunate feature that if one takes them at face value, many quantities of interest appear to be infinite! Various techniques, usually going under the common name of “renormalisation” have been developed over the years to address this, allowing mathematicians and physicists to tame these infinities. We will tip our toes into some of the mathematical aspects of these techniques and we will see how they have recently been used to make precise analytical statements about the solutions of some equations whose meaning was not even clear until now.

Lectures 2 and 3: The BPHZ theorem for stochastic PDEs

Tuesday, October 25, 2016, 4:30pm–5:30pm, Eckhart 202

Wednesday, October 26, 2016, 4pm–5pm, Eckhart 202

Abstract: The Bogoliubov-Parasiuk-Hepp-Zimmermann theorem is a cornerstone of perturbative quantum field theory: it provides a consistent way of "renormalising" the diverging integrals appearing there to turn them into bona fide distributions. Although the original article by Bogoliubov and Parasiuk goes back to the late 50s, it took about four decades for it to be fully understood. In the first lecture, we will formulate the BPHZ theorem as a purely analytic question and show how its solution arises very naturally from purely algebraic considerations. In the second lecture, we will show how a very similar structure arises in the context of singular stochastic PDEs and we will present some very recent progress on its understanding, both from the algebraic and the analytical point of view.

• Charles Amick Memorial Lectures by Jennifery Chayes -Microsoft Resesarch.

First Lecture : Modeling and Estimating Massive Networks: Overview

October 28, 4PM, Ryerson 251

Second Lecture: Limits and Stochastic Models for Sparse Massive Networks

October 31, 4PM, Eckhart 202

Third Lecture: Exchangeablity and Estimation of Sparse Massive Networks

November 1, 4PM, Eckhart 206

• Friday Nov 4: Ramon van Handel -Princeton University.

Title: The Borell-Ehrhard game

Abstract: A precise description of the convexity of Gaussian measures is provided by a remarkable Brunn-Minkowski type inequality due to Ehrhard and Borell. The delicate nature of this inequality has complicated efforts to develop more general geometric inequalities in Gauss space that mirror the rich family of results in the classical Brunn-Minkowski theory. In this talk, I will aim to shed some new light on Ehrhard's inequality by showing that it arises from a somewhat unexpected game-theoretic mechanism. This insight makes it possible to identify new results, such as an improved form of Barthe's reverse Brascamp-Lieb inequality in Gauss space. If time permits, I will also outline how probabilistic ideas enabled us (in work with Yair Shenfeld) to settle the equality cases in the Ehrhard-Borell inequalities.

• Friday Nov 18: Xinyi Li -University of Chicago.

Title: Percolative properties of Brownian interlacements and its vacant set

Abstract: In this talk, I will give a brief introduction to Brownian interlacements, and investigate various percolative properties regarding this model. Roughly speaking, Brownian interlacements can be described as a certain Poissonian cloud of doubly-infinite continuous trajectories in the d-dimensional Euclidean space, d greater or equal to 3, with the intensity measure governed by a level parameter. We are interested in both the interlacement set, which is an enlargement (“the sausages”) of the union of the trace in the aforementioned cloud of trajectories, and the vacant set, which is the complement of the interlacement set. I will talk about the following results: 1) The interlacement set is “well-connected”, i.e., any two “sausages” in d-dimensional Brownian interlacements, can be connected via no more than ceiling((d − 4)/2) intermediate sausages almost surely. 2) The vacant set undergoes a non-trivial percolation phase transition when the level parameter varies.

## Spring 2016 Seminars

• Friday, April 1: Stephane Benoist -Columbia University.

Title: Conformally invariant loop measures

Abstract: We will discuss several aspects of a conjecture by Kontsevich and Suhov regarding existence and uniqueness of a one parameter family of conformally invariant measures on simple loops (conjecturally related to the SLE family). The most natural case (zero central charge i.e. SLE parameter kappa=8/3) was understood in a paper of Werner predating the conjecture. In a work in progress, Dubédat and myself construct loop measures in the whole conjectural range of existence (i.e. parameters kappa for which SLE is a simple curve).

• Friday, April 8: Krzysztof Burdzy -University of Washington.

Title: Twin peaks

Abstract: I will discuss some questions and results on random labelings of graphs conditioned on having a small number of peaks (local maxima). The main open question is to estimate the distance between two peaks on a large discrete torus, assuming that the random labeling is conditioned on having exactly two peaks. Joint work with Sara Billey, Soumik Pal, Lerna Pehlivan and Bruce Sagan.

• Friday, April 15: Hao Shen -Columbia University.

Title: Regularity structure theory and its applications

Abstract: Stochastic PDEs arise as important models in probability and mathematical physics. They are typically nonlinear, driven by very singular random forces. Due to lack of regularity it is typically very challenging to even interpret what one means by a solution. In this talk I will explain the solution theories for some of these equations, with a focus on the theory of regularity structures recently developed by Martin Hairer. As applications of these theories, one can make sense of the solutions to these stochastic PDEs, and once their solution theories are established various convergence or approximation problems can be tackled.

• Friday, April 29: Alex Dunlap - Stanford University.

Title: First passage percolation on the exponential of two-dimensional branching random walk: subsequential scaling limit at high temperature

Abstract: Abstract: Let \{\eta_{N, v}: v\in V_N\} be a branching random walk in a two-dimensional box V_N of side length N, that is, a 4-ary BRW with Gaussian increments indexed by lattice points (with approximately log-correlated covariances). We study the first passage percolation metric where each vertex v is given a random weight of e^{\gamma \eta_{N, v}}. I will show that for sufficiently small but fixed \gamma>0, for any sequence of \{N_k\} there exists a subsequence along which the appropriately scaled FPP metric converges in the Gromov-Hausdorff sense to a random metric on the unit square in R^2. In addition, all possible (conjecturally unique) scaling limits are non-trivial and are continuous with respect to the Euclidean metric. Joint work with J. Ding.

• Friday, May 6: no seminar. Conference on New Developments in Probability@ Northwestern University.

• Friday, May 13: no seminar. Statistics Department anniversarial conference.

• Friday, May 20: Jun Yin -University of Wisconsin.

Title: Delocalization and Universality of band matrices

Abstract: In this talk we introduce our new work on band matrices, whose eigenvectors and eigenvalues are widely believed to have the same asymptotic behaviors as those of Wigner matrices. We proved that this belief is true as long as the bandwidth is wide enough.

• Friday, May 27: no seminar. Workshop on percolation, spin glasses and random media@ Northwestern University.

## Winter 2016 Seminars

• Friday, Jan 8: Tom Hutchcroft - University of British Columbia.

Title: Circle packing and uniform spanning forests of planar graphs

Abstract: The Koebe-Andreev-Thurston Circle Packing Theorem lets us draw planar graphs in a canonical way, so that the geometry of the drawing reveals analytic properties of the graph. Circle packing has proven particularly effective in the study of random walks on planar graphs, where it allows us to estimate various quantities in terms of their counterparts for Brownian motion in the plane. In this talk, I will introduce the theory of circle packing and discuss work with Asaf Nachmias in which we use circle packing to study uniform spanning forests of planar graphs, a probability model closely related to random walk. We prove that the free uniform spanning forest of any bounded degree, proper planar graph is connected almost surely, answering positively a question of Benjamini, Lyons, Peres and Schramm. Our proof is quantitative, and also shows that uniform spanning forests exhibit some of the same behaviour universally for all bounded degree transient triangulations, provided that one measures distances and areas in the triangulation using the hyperbolic geometry of its circle packing rather than with the usual graph metric and counting measure.

• Friday, Jan 15: Ewain Gwynne - MIT.

Title: An almost sure KPZ relation for SLE and Brownian motion

Abstract: I will discuss a KPZ-type formula which relates the Hausdorff dimension of any set associated with SLE, CLE, or related processes; and the Hausdorff dimension of a corresponding set associated with a correlated two-dimensional Brownian motion. In many cases, the dimension of the Brownian motion set is already known or easy to compute. This gives rise to new proofs of the dimensions of several sets associated with SLE, including the SLE curve; the double points and cut points of SLE; and the intersection of two flow lines of a Gaussian free field. The formula is based on the peanosphere construction of Duplantier, Miller, and Sheffield (2014), which encodes a Liouville quantum gravity (LQG) surface decorated with an independent space-filling SLE curve by means of a correlated two-dimensional Brownian motion. I will give a moderately detailed overview of this construction. Based on a joint work with Nina Holden and Jason Miller http://arxiv.org/abs/1512.01223.

• Friday, Jan 22: Aukosh Jagannath - NYU.

Title: The Parisi variational problem

Abstract: The Parisi Variational Problem is a challenging non-local, strictly convex variational problem over the space of probability measures whose analysis is of great interest to the study of mean field spin glasses. In this talk, I present a conceptually simple approach to the study of this problem using techniques from PDEs, stochastic optimal control, and convex optimization. We begin with a new characterization of the minimizers of this problem whose origin lies in the first order optimality conditions for this functional. As a demonstration of the power of this approach, we study a prediction of de Almeida and Thouless regarding the validity of the 1 atomic anzatz. We generalize their conjecture to all mixed p-spin glasses and prove that their condition is correct in the entire temperature-external field plane except for a compact set whose phase is unknown at this level of generality. A key element of this analysis is a new class of estimates regarding gaussian integrals in the large noise limit called Dispersive Estimates of Gaussians’’ . This is joint work with Ian Tobasco (NYU Courant).

• Friday, Jan 29: Hao Wu - Universite de Geneve.

Title: Arm Exponents for SLE

Abstract: In the study of lattice models, arm exponents play an important role. In this talk, we first discuss the arm exponents for critical percolation, explain how they are derived and why they are important. Second, we introduce the arm exponents for chordal SLE and explain the application to the critical Ising and FK-Ising model. Finally, we give a brief idea on deriving these exponents and some related open questions.

• Friday, Feb 12: Greg Lawler - University of Chicago.

Title: Convergence of naturally parametrized loop-erased random walk to the Schramm-Loewner evolution parametrized by Minkowski content

Abstract: The main goal of this talk is to explain the title. I will define the terms (type of convergence, naturally parametrized, loop-erased random walk, Schramm-Loewner evolution, Minkowski content) as well as the result. This is based on work with Fredrik Wiklund.

• Friday, Feb 19: Robin Pemantle - University of Pennsylvania.

Title: Evolution of one-cells on a line

Abstract: We consider systems with the following description. At time zero, the real line is partitioned into intervals. The original partition, which may be random, evolves according to a deterministic rule whereby the interface between consecutive pair of cells move so that the larger cell grows and the smaller cell shrinks. When a cell shrinks to zero it disappears and the two bounding points coalesce. I will discuss one such system: a somewhat degenerate one-dimensional version of a two (and higher) dimensional mean-curvature flow model about which almost nothing rigorous is known. In joint work with Emanuel Lazar, we prove that the Poisson measure is invariant for this evolution, provided that space is rescaled exponentially. We do this by introducing the dual process (time-reversal). This process, unlike the forward process, contains some randomness and may be exactly analyzed. A number of questions remain open, such as uniqueness of trajectories, convergence to Poisson from other initial conditions, and stability under perturbation. Finally, I will discuss other one-dimensional models with similar descriptions about which even less is known.

• Cancelled! Friday, March 4: Brian Rider -Temple University.

## Fall 2015 Seminars

• Friday, Oct 2: Elchanan mossel - University of Pennsylvania and U.C. Berkeley.

Title: Correlation distillation in probability spaces

Abstract: Given a finite exchangeable collection of random variables in a probability space, the correlation distillation problem asks for the partition of the space into sets of a given measure as to maximize the probability that all random variables lie in the same set. This problem is closely related to isoperimetric problems and is motivated by applications in voting, theoretical computer science and information theory. In the talk I will survey some older and some recent results on correlation distillation. Many open problems will be presented.

• (Local probabiltiy events) Oct 3-4, AMS meeting in probability at Loyola .

• (No Seminar!) Oct 8-10, Midwest Probability Colloquium at Northwestern University.

• Friday, Oct 16: Jelani Nelson - Harvard University.

Title: Dimensionality Reduction Via Sparse Matrices

Abstract: This talk will discuss sparse Johnson-Lindenstrauss transforms, i.e. sparse linear maps into much lower dimension which preserve the Euclidean geometry of a set of vectors. Both upper and lower bounds will be presented, as well as applications to certain domains such as numerical linear algebra and compressed sensing. Based on various joint works with Jean Bourgain, Sjoerd Dirksen, Daniel M. Kane, and Huy Le Nguyen.

• Friday, Oct 23: Tianyi Zheng - Stanford University.

Title: Speed of random walks on Cayley graphs of finitely generated groups

Abstract: In this walk I will discuss a new construction of a family of groups. We show that up to an absolute constant factor, any function $f$ satisfying $f(1)=1$, $f(n)/\sqrt{n}$, $n/f(n)$ both non-decreasing can be realized as speed function of simple random walk on some finitely generated group. In particular, it implies any number in [1/2,1] can be realized as the speed exponent of simple random walk on some group. The construction is very flexible and allows us to answer positively a recent conjecture of Gideon Amir regarding joint behavior of speed and entropy. We evaluate the Hilbert compression exponents of the groups under consideration. In particular, we show that for any $\alpha\in[2/3,1]$, there exists a 3-step solvable group with Hilbert compression exponent $\alpha$. It follows that there exists uncountably many pairwise non quasi-isometric finitely generated 3-step solvable groups. Joint work with Jeremie Brieussel.

• Friday, Nov 6: Charles Smart - University of Chicago.

Title: SPDE techniques for the random conductance model

Abstract: I will survey some of the recent work applying techniques from partial differential equations to the random conductance model on the lattice. This will include some work of mine with Armstrong and some work of Armstrong-Kuusi-Mourat and Gloria-Otto. There are now two approaches to obtaining optimal rates in stochastic homogenization in divergence form. The first obtains Green's function estimates by appealing to the Efron-Stein concentration inequality. The second uses regularity theory to localize the dependence of the solution on the coefficients. I will discuss both of these methods.

• Friday, Nov 13: Xin Sun - MIT.

Title: Almost Sure Multifractal Spectrum of SLE

Abstract: 15 years ago B. Duplantier predicted the multifractal spectrum of Schramm Loewner Evolution (SLE), which encodes the fine structure of the harmonic measure of SLE curves. In this talk, I will report our recent rigorous derivation of this prediction. As a byproduct, we also confirm a conjecture of Beliaev and Smirnov for the a.s. bulk integral means spectrum of SLE. The proof uses various couplings of SLE and Gaussian free field, which are developed in the theory of imaginary geometry and Liouville quantum gravity. (Joint work with E. Gwynne and J. Miller.)

• Friday, Nov 20: Rodrigo Bañuelos - Purdue University.

Title: The Hardy-Littlewood-Sobolev inequality via martingale transforms

Abstract: We outline a martingale proof of the classical Hardy-Littlewood-Sobolev (HLS) inequality which naturally extends to the setting of Markovian semigroups that have finite dimension in the sense of Varopoulos. The motivation for this approach comes from efforts to employ probabilistic techniques to study (extend) the sharp HLS inequality of E.H.Lieb.

• Friday, Dec 4: Mykhaylo Shkolnikov - Princeton Univeristy.

Title: On multilevel Dyson Brownian motions

Abstract: I will discuss how Dyson Brownian motions describing the evolution of eigenvalues of random matrices can be extended to multilevel Dyson Brownian motions describing the evolution of eigenvalues of minors of random matrices. The construction is based on intertwining relations satisfied by the generators of Dyson Brownian motions of different dimensions. Such results allow to connect general beta random matrix theory to particle systems with local interactions, and to obtain novel results even in the case of classical GOE, GUE and GSE random matrix models. Based on joint work with Vadim Gorin.

• Friday, Dec 11: Stefan Adams - University of Warwick.

Title: Isomorphism theorems for space-time random walks

Abstract: Loop measures have become important in the analysis of random walks and connected research in mathematical physics. Such measures go back to Symanzik in the late 1960s in the context of Euclidean field theory. We discuss loop measures on graphs with countable infinite different time horizons. These measures are connected to the cycle representation of partition functions in quantum systems (Boson systems). We derive corresponding Dynkin isomorphism theorems for space-time random walks and we prove for some specific models the onset of the so-called Bose-Einstein condensation.

## Spring 2015 Seminars

• Friday, April 3rd: Yimin Xiao - Michigan State University.

Title: Discrete Fractal Dimensions and Large Scale Multifractals

Abstract: Ordinary fractal dimensions such as Hausdorff dimension and packing dimension are useful for analyzing the (microscopic) geometric structures of various thin sets and measures. For studying (macroscopic or global) fractal phenomena of discrete sets, Barlow and Taylor (1989, 1992) introduced the notions of discrete Hausdorff and packing dimensions. In this talk we present some recent results on macroscopic multifractal properties of random sets associated with the Ornstein-Uhlenbeck process and the mild solution of the parabolic Anderson model. (Joint work with Davar Khoshnevisan and Kunwoo Kim.)

• (Cancelled!) Friday, April 10th: Mykhaylo Shkolnikov - Princeton Univeristy.

Title: On multilevel Dyson Brownian motions.

Abstract: I will discuss how Dyson Brownian motions describing the evolution of eigenvalues of random matrices can be extended to multilevel Dyson Brownian motions describing the evolution of eigenvalues of minors of random matrices. The construction is based on intertwining relations satisfied by the generators of Dyson Brownian motions of different dimensions. Such results allow to connect general beta random matrix theory to particle systems with local interactions, and to obtain novel results even in the case of classical GOE, GUE and GSE random matrix models. Based on joint work with Vadim Gorin.

• (Speical time and location!) Thursday, April 16th, Billingsley Lecture by Wendelin Werner -ETH.

• Friday, April 17th: Wendelin Werner -ETH.

Title: A simple renormalization flow setup for FK-percolation models

Abstract: We will present a simple setup in which one can make sense of a renormalization flow for FK-percolation models in terms of a simple Markov process on a state-sace of discrete weighted graphs. We will describe how to formulate the universality conjectures in this framework (in terms of stationary measures for this Markov process), and how to prove this statement in the very special case of the two-dimensional uniform spanning tree (building on existing results on this model). This is based in part on joint work with Stéphane Benoist and Laure Dumaz.

• Friday, April 24th: Soumik Pal -University of Washington.

Title: Dynamics on random regular graphs: Dyson Brownian motion and the Poisson free field

Abstract: : A single permutation, seen as union of disjoint cycles, represents a regular graph of degree two. Consider d many independent random permutations and superimpose their graph structures. It is a common model of a random regular (multi-) graph of degree 2d. Consider the problem of eigenvalue fluctuations of the adjacency matrix of such a graph. We consider the following dynamics. The ‘dimension’ of each permutation grows by coupled Chinese Restaurant Processes, while in ‘time’ each permutation evolves according to the random transposition Markov chain. Asymptotically in the size of the graph one observes a remarkable evolution of short cycles and linear eigenvalue statistics in dimension and time. We give a Poisson random surface description in dimension and time of the limiting cycle counts for every d. As d grows to infinity, these Poisson random surfaces converge to the Gaussian Free Field preserved in time by the Dyson Brownian motion. Part of this talk is based on a joint work with Tobias Johnson and the rest is based on a joint work with Shirshendu Ganguly. (Cambridge).

• Friday, May 1st: Louis-Pierre Arguin -University of Montreal.

Title: Maxima of log-correlated Gaussian fields and of the Riemann Zeta function

Abstract: A recent conjecture of Fyodorov, Hiary & Keating states that the maxima of the Riemann Zeta function on a bounded interval of the critical line behave similarly to the maxima of a specific class of Gaussian fields, the so-called log-correlated Gaussian fields. These include important examples such as branching Brownian motion and the 2D Gaussian free field. In this talk, we will highlight the connections between the number theory problem and the probabilistic models. We will outline the proof of the conjecture in the case of a randomized model of the Zeta function. We will discuss possible approaches to the problem for the function itself. This is joint work with D. Belius (NYU) and A. Harper (Cambridge).

• Friday, May 8th: Van Vu -Yale University.

Title: Random matrices have simple spectrum

Abstract: A symmetric matrix has simple spectrum if all eigenvalues are different. Babai conjectured that random graphs have simple spectrum with probability tending to 1. Confirming this conjecture, we prove the simple spectrum property for a large class of random matrices. If time allows, we will discuss the harder problem of bounding the spacings between consecutive eigenvalues, with applications in mathematical physics, computer science, and numerical linear algebra. Several open questions will also be presented. Joint work with H. Nguyen (OSU) and T. Tao (UCLA).

• Friday, May 15th: Pierluigi Contucci -U. Bologna.

Title: Exactly solvable mean-field monomer-dimers models

Abstract: A The seminar will introduce some mean-field models used to describe monomer-dimers systems. In particular the solution for the diluted case and the random impurity case will be shown and the absence of phase transition proved.

• Friday, May 22th: Dan Romik -UC Davis.

Title: A local central limit theorem for random representations of SU(3)

Abstract: The number p(n) of integer partitions of n is given approximately for large n by a famous asymptotic formula proved by Hardy and Ramanujan in 1918. This can be interpreted as a statement about the number of inequivalent representations of dimension n of the group SU(2). In this talk I will discuss my recent proof of an analogous result for the asymptotic number of n-dimensional representations of the group SU(3). A key step is to prove a local central limit theorem in a probabilistic model for random representations, which requires some ideas from the theory of modular forms. I will explain these ideas, as well as connections to a mysterious “Witten zeta function" associated with SU(3), and additional applications to understanding the limit shape of random n-dimensional representations of SU(3). No knowledge of representation theory will be assumed or needed.

• Friday, May 29th: Eyal Lubetzky -NYU.

Title: Effect of initial conditions on mixing for the Ising Model

Abstract: Recently, the information percolation'' framework was introduced as a way to obtain sharp estimates on mixing for the high temperature Ising model, and in particular, to establish cutoff in three dimensions up to criticality from a worst starting state. I will describe how this method can be used to understand the effect of different initial states on the mixing time, both random (''warm start'') and deterministic. Joint work with Allan Sly.

• (Double talk!) 2:30-3:30 Friday, June 5th: Maury Bramson -UMN.

Title: Proportional Switching in FIFO Networks

Abstract: A central problem in queueing theory is the development of policies that efficiently allocate available resources. Many standard policies have a fixed capacity at individual sites, rather than the ability to allocate resources across sites. We discuss here the proportional switching policy, where the amount of service at different sites is dependent and the corresponding service vector is required to lie in a convex region. We also assume that packets are served in the FIFO (first-in, first-out) order. Past work on the stability of proportional switching networks has focused on networks with elementary routing structure (such as immediate departure after service at a site). Here, we consider the stability problem for general routing structures. The talk is based on joint work with B. D'Auria and N. Walton.

3:35-4:35 Friday, June 5th: Paul Jung -University of Alabama Birmingham.

Title: Levy Khintchine random matrices and the Poisson weighted infinite skeleton tree

Abstract: We study a class of Hermitian random matrices which includes Wigner matrices, heavy-tailed random matrices, and sparse random matrices such as adjacency matrices of Erdos-Renyi graphs with p=1/N. Our matrices have real entries which are i.i.d. up to symmetry. The distribution of entries depends on N, and we require sums of rows to converge in distribution; it is then well-known that the limit must be infinitely divisible. We show that a limiting empirical spectral distribution (LSD) exists, and via local weak convergence of associated graphs, the LSD corresponds to the spectral measure associated to the root of a graph which is formed by connecting infinitely many Poisson weighted infinite trees using a backbone structure of special edges. One example covered are matrices with i.i.d. entries having infinite second moments, but normalized to be in the Gaussian domain of attraction. In this case, the LSD is a semi-circle law.

## Winter 2015 Seminars

• Friday, January 9th: Sebastien Roch - UW-Madison.

Title: Recent results on the multispecies coalescent

Abstract: The multispecies coalescent is a variant of Kingman’s coalescent in which several populations are stitched together on a base tree. Increasingly, it plays an important role in phylogenetics where it can be used to model the joint evolution of a large number of genes across multiple species. Motivated by information-theoretic questions, I will present a recent probabilistic analysis of the multispecies coalescent which establishes fundamental limits on the inference of this model from molecular sequence data. No biology background is required. This is joint work with Gautam Dasarathy, Elchanan Mossel, Rob Nowak, and Mike Steel.

• Friday, January 16th: Steve Lalley - Univ. Chicago.

Title: Nash Equilibria for a Quadratic Voting Game

Abstract: Voters making a binary decision purchase votes from a centralized clearing house, paying the square of the number of votes purchased. The net payoff to an agent with utility u who purchases v votes is \Psi(S)u−v^2, where \Psi is an odd, monotone function taking values between -1 and +1 and S is the sum of all votes purchased by the n voters participating in the election. The utilities of the voters are assumed to arise by random sampling from a probability distribution F with compact support; each voter knows her own utility, but not those of the other voters, although she does know the sampling distribution F. Nash equilibria for this game are described.

• Friday, January 23th: Shuwen Lou - UIC.

Title: Brownian motion on spaces with varying dimension

Abstract: The model can be picturized as the random movement of an insect on the ground with a pole standing on it. That is, part of the state space has dimension 2, and the other part of the state space has dimension 1. We define such a process as a darning process'' in terms of Dirichlet form, because 2-dimensional Brownian motion does not hit any singleton. We show that the behavior of this process switches between 1-dimensional and 2-dimensional, which depends on both the time and the positions of the points. An open ongoing project will also be introduced: Can we approximate such a process by random walks? The main results of this talk are based on my joint work with Zhen-Qing Chen.

• Friday, February 6th: Yury Makarychev - TTIC.

Title: Constant Fac­tor Approx­i­ma­tion for Bal­anced Cut in the PIE Model

Abstract: We pro­pose and study a new semi-random semi-adversarial model for Bal­anced Cut, a planted model with permutation-invariant ran­dom edges (PIE). Our model is much more general than planted and stochastic mod­els con­sid­ered pre­vi­ously. Con­sider a set of ver­tices V par­ti­tioned into two clus­ters L and R of equal size. Let G be an arbi­trary graph on V with no edges between L and R. Let E_random be a set of edges sam­pled from an arbi­trary permutation-invariant dis­tri­b­u­tion (a dis­tri­b­u­tion that is invari­ant under per­mu­ta­tion of ver­tices in L and in R). Then we say that G + E_random is a graph with permutation-invariant ran­dom edges. We present an approx­i­ma­tion algo­rithm for the Bal­anced Cut prob­lem that finds a bal­anced cut of cost O(|E_random|) + n polylog(n) in this model. In the regime when there are at least \Omega(n polylog(n)) random edges, this is a con­stant fac­tor approx­i­ma­tion with respect to the cost of the planted cut. Joint work with: Konstantin Makarychev and Aravin­dan Vijayaraghavan.

• Friday, February 13th: James R. Lee - University of Washington.

Title: Regularization under diffusion and Talagrand's convolution conjecture

Abstract: It is a well-known phenomenon that functions on Gaussian space become smoother under the Ornstein-Uhlenbeck semigroup. For instance, Nelson's hypercontractive inequality shows that if p > 1, then L^p functions are sent to L^q functions for some q > p. In 1989, Talagrand conjectured* that quantitative smoothing is achieved even for functions which are only L^1, in the sense that under the semigroup, such functions have tails that are strictly better than those predicted by Markov's inequality and preservation of mass. Ball, Barthe, Bednorz, Oleszkiewicz, and Wolff (2010) proved that this holds in fixed dimensions. We resolve Talagrand's conjecture conjecture positively (with no dimension dependence). The key insight is to study a subset of Gaussian space at various granularities by approaching it as "efficiently" as possible. To this end, we employ an Ito process that arose in the context of optimal control theory. Efficiency is measured by the average "work" required to couple the approach process to a Brownian motion. *Talagrand's full conjecture is for functions on the discrete cube. Here we address the Gaussian limiting case. This is joint work with Ronen Eldan.

• Friday, February 20th: Philippe Sosoe - Harvard.

Title: On the chemical distance in critical percolation

Abstract: In two-dimensional critical percolation, the works of Aizenman-Burchard and Kesten-Zhang imply that macroscopic distances inside percolation clusters are bounded below by a power of the Euclidean distance greater than 1+ε, for some positive ε. No more precise lower bound has been given so far. Conditional on the existence of an open crossing of a box of side length n, there is a distinguished open path which can be characterized in terms of arm exponents: the lowest open path crossing the box. This clearly gives an upper bound for the shortest path. The lowest crossing was shown by Zhang and Morrow to have volume n^4/3+o(1) on the triangular lattice. Following a question of Kesten and Zhang, we compare the length of shortest circuit in an annulus to that of the innermost circuit (defined analogously to the lowest crossing). I will explain how to show that the ratio of the expected length of the shortest circuit to the expected length of the innermost crossing tends to zero as the size of the annulus grows. Joint work with Jack Hanson and Michael Damron.

• Friday, February 27th: Partha Dey - UIUC.

Title: High temperature limits for $(1+1)$-d directed polymer with heavy-tailed disorder.

Abstract: The directed polymer model at intermediate disorder regime was introduced by Alberts-Khanin-Quastel (2012). It was proved that at inverse temperature $\beta n^{-\gamma}$ with $\gamma=1/4$ the partition function, centered appropriately, converges in distribution and the limit is given in terms of the solution of the stochastic heat equation. This result was obtained under the assumption that the disorder variables posses exponential moments, but its universality was also conjectured under the assumption of six moments. We show that this conjecture is valid and we further extend it by exhibiting classes of different universal limiting behaviors in the case of less than six moments. We also explain the behavior of the scaling exponent for the log-partition function under different moment assumptions and values of $\gamma$. Based on joint work with Nikos Zygouras.

• Friday, March 6th: Renming Song - UIUC.

Title: Stochastic flows for Levy processes with Holder drifts

Abstract: In this talk I will present some new results on the following SDE in $R^d$: $$dX_t=b(t, X_t)dt+dZ_t, \quad X_0=x,$$ where $Z$ is a Levy process. We show that for a large class of Levy processes $Z$ and Holder continuous drfit $b$, the SDE above has a unique strong solution for every starting point $x\in R^d$. Moreover, these strong solutions form a $C^1$-stochastic flow. In particular, we show that, when $Z$ is a symmetric $\alpha$-stable process with $\alpha\in (0, 1]$ and $b$ is $\beta$-Holder continuous with $\beta\in (1-\alpha/2, 1)$, the SDE above has a unique strong solution.

• Friday, March 13th: Wei-Kuo Chen - Univ. Chicago.

Title: Universality in spin glasses

Abstract: This talk is concerned about some universal properties of the Parisi solution in spin glass models. We will show universality of chaos phenomena and ultrametricity in the mixed p-spin model under mild moment assumptions on the environment. We will explain that the results also extend to quenched self-averaging of some physical observables in the mixed p-spin model as well as in different spin glass models including the Edwards-Anderson model and the random field Ising model.

## Fall 2014 Seminars

• Friday, Oct 3rd (Speical time: 1:30-2:30!): Prasad Tetali - Georgia Institute of Technology.

Title: Displacement convexity of entropy and curvature in discrete settings

Abstract: Inspired by exciting developments in optimal transport and Riemannian geometry (due to the work of Lott-Villani and Sturm), several independent groups have formulated a (discrete) notion of curvature in graphs and finite Markov chains. I will describe some of these approaches briefly, and mention some related open problems of potential independent interest.

• Friday, Oct 10th: No seminar. Midwest Probability Colloqium

• Friday, Oct 17th: Thomas Liggett - UCLA.

Title: Finitely Dependent Coloring on Z and other Graphs

Abstract: In 2008, Oded Schramm asked the following question: For what values of $k$ and $q$ does there exist a stationary, proper, $k-$dependent $q-$coloring of the integers? Schramm had a substantial amount of evidence, which I will describe, that convinced him that such a coloring does not exist for any values of $k$ and $q$. In fact, it turns out that such an object does exist for many values of $k$ and $q$. I will tell you exactly which ones work, and will describe colorings with these properties. No knowledge of advanced probability is needed to follow the lecture. There are several connections with combinatorics, but again, no specialized knowledge is needed. This is joint work with A. Holroyd.

• Friday, Oct 24th: Tonći Antunović - UCLA.

Title: Stationary Eden Model on amenable groups

Abstract: We consider stationary versions of the Eden model, on a product of a Cayley graph G of an amenable group and positive integers. The process results in a collection of disjoint trees rooted at G, each of which consists of geodesic paths in a corresponding first passage percolation model on the product graph. Under weak assumptions on the weight distribution and by relying on ergodic theorems, we prove that almost surely all trees are finite. This generalizes certain known results on the two-type Richardson model, in particular of Deijfen and Haggstrom on the Euclidean lattice. This is a joint work with Eviatar Procaccia.

• Friday, Nov 7th: Jonathan Novak - MIT.

Title: Random tilings and Hurwitz numbers

Abstract: This talk is about random tilings of a special class of planar domains, which I like to call "sawtooth domains." Sawtooth domains have the special feature that their tilings are in bijective correspondence with Gelfand-Tsetlin patterns, aka semistandard Young tableaux. Consequently, many observables can be expressed in terms of special functions of representation-theoretic origin. In particular, the distribution of tiles of one type along a horizontal slice through a uniformly random tiling is encoded by the Harish-Chandra/Itzykson-Zuber integral, a familiar object from random matrix theory which also happens to be a generating function for a desymmetrized version of the Hurwitz numbers from enumerative algebraic geometry. I will explain how this fact allows one to prove that tiles along a slice fluctuate like the eigenvalues of a Gaussian random matrix.

• Friday, Nov 14th: Nayantara Bhatnagar - University of Delaware.

Title: Lengths of Monotone Subsequences in a Mallows Permutation

Abstract: The longest increasing subsequence (LIS) of a uniformly random permutation is a well studied problem. Vershik-Kerov and Logan-Shepp first showed that asymptotically the typical length of the LIS is 2sqrt(n). This line of research culminated in the work of Baik-Deift-Johansson who related this length to the Tracy-Widom distribution. We study the length of the LIS and LDS of random permutations drawn from the Mallows measure, introduced by Mallows in connection with ranking problems in statistics. Under this measure, the probability of a permutation p in S_n is proportional to q^Inv(p) where q is a real parameter and Inv(p) is the number of inversions in p. We determine the typical order of magnitude of the LIS and LDS, large deviation bounds for these lengths and a law of large numbers for the LIS for various regimes of the parameter q. This is joint work with Ron Peled.

• Friday, Nov 21th: Antonio Auffinger - Northwestern University.

Title: Rate of convergence of the mean for sub-additive ergodic sequences

Abstract: For a subadditive ergodic sequence {X_{m,n}}, Kingman's theorem gives convergence for the terms X_{0,n}/n to some non-random number g. In this talk, I will discuss the convergence rate of the mean EX_{0,n}/n to g. This rate turns out to be related to the size of the random fluctuations of X_{0,n}; that is, the variance of X_{0,n}, and the main theorems I will present give a lower bound on the convergence rate in terms of a variance exponent. The main assumptions are that the sequence is not diffusive (the variance does not grow linearly) and that it has a weak dependence structure. Various examples, including first and last passage percolation, bin packing, and longest common subsequence fall into this class. This is joint work with Michael Damron and Jack Hanson.

• Friday, Dec 5th: Brent M. Werness -University of Washington.

Title: Hierarchical approximations to the Gaussian free field and fast simulation of Schramm-Loewner evolutions

Abstract: The Schramm--Loewner evolutions (SLE) are a family of stochastic processes which describe the scaling limits of curves which occur in two-dimensional critical statistical physics models. SLEs have had found great success in this task, greatly enhancing our understanding of the geometry of these curves. Despite this, it is rather difficult to produce large, high-fidelity simulations of the process due to the significant correlation between segments of the simulated curve. The standard simulation method works by discretizing the construction of SLE through the Loewner ODE which provides a quadratic time algorithm in the length of the curve. Recent work of Sheffield and Miller has provided an alternate description of SLE, where the curve generated is taken to be a flow line of the vector field obtained by exponentiating a Gaussian free field. In this talk, I will describe a new hierarchical method of approximately sampling a Gaussian free field, and show how this allows us to more efficiently simulate an SLE curve. Additionally, we will briefly discuss questions of the computational complexity of simulating SLE which arise naturally from this work.

## Spring 2014 Seminars

• Friday, April 18th: Shankar Bhamidi - UNC.

Title: Limited choice and randomness in the evolution of networks

Abstract: The last few years have seen an explosion in network models describing the evolution of real world networks. In the context of math probability, one aspect which has seen an intense focus is the interplay between randomness and limited choice in the evolution of networks, ranging from the description of the emergence of the giant component, the new phenomenon of explosive percolation'' and power of two choices. I will describe ongoing work in understanding such dynamic network models, their connections to classical constructs such as the standard multiplicative coalescent and applications of these simple models in fitting retweet networks in Twitter.

• Friday, May 2nd: Tai Melcher - University of Virginia.

Title: An example of hypoellipticity in infinite dimensions

Abstract: A collection of vector fields on a manifold satisfies H\"{o}rmander's condition if any two points are connected by a path whose tangent vectors only lie in the given directions. It is well-known that a diffusion which is allowed to travel only in these directions is smooth, in the sense that its transition probability measure is absolutely continuous with respect to the volume measure and has a strictly positive smooth density. Smoothness results of this kind in infinite dimensions are typically not known, the first obstruction being the lack of an infinite-dimensional volume measure. We will discuss recent results on a particular class of infinite-dimensional spaces, where we have shown that vector fields satisfying H\"{o}rmander's condition generate a diffusion which has a strictly positive smooth density with respect to an appropriate reference measure.

• Wednesday, May 7th: Alice Guionnet -MIT.

(Mathematics Colloquium, 2:00 -3:00 pm @ Eckhart 206 )

Title: Free probability and random matrices; from isomorphisms to universality

Abstract: Free probability is a probability theory for non-commutative variables introduced by Voiculescu about thirty years ago. It is equipped with a notion of freeness very similar to independence. It is a natural framework to study the limit of random matrices with size going to infinity. In this talk, we will discuss these connections and how they can be used to adapt ideas from classical probability theory to operator algebra and random matrices. We will in particular focus on how to adapt classical ideas on transport maps following Monge and Ampere to construct isomorphisms between algebras and prove universality in matrix models. This talk is based on joint works with F. Bekerman, Y. Dabrowski, A. Figalli and D. Shlyakhtenko.

• Friday, May 9th: Antonio Auffinger - University of Chicago.

Title: Strict Convexity of the Parisi Functional

Abstract: Spin glasses are magnetic systems exhibiting both quenched disorder and frustration, and have often been cited as examples of "complex systems." As mathematical objects, they provide several fascinating structures and conjectures. This talk will cover recent progress that shed more light in the mysterious and beautiful solution proposed 30 years ago by G. Parisi. We will focus on properties of the free energy of the famous Sherrington-Kirkpatrick model and we will explain a recent proof of the strict convexity of the Parisi functional. Based on a joint work with Wei-Kuo Chen.

• Friday, May 16th -- Double Talk! :

(2:30-3:30) Elton P. Hsu - Northwestern University.

Title: Brownian Motion and Gradient Estimates of Positive Harmonic Functions

Abstract: Many gradient estimates in differential geometry can be naturally treated by stochastic methods involving Brownian motion on a Riemannian manifold. In this talk, we discuss Hamilton\'92s gradient estimate of bounding the gradient of the logarithm of a positive harmonic function in terms of its supremum from this point of view. We will see how naturally this gradient estimate follows from Ito\'92s formula and extend it to manifolds with boundary by considering reflecting Brownian motion. Furthermore, we will show that in fact Hamilton\'92s gradient estimate can be embedded as the terminal case of a family of gradient estimates which can be treated just as easily by the same stochastic method.

(4:00-5:00) Marek Biskup -UCLA.

Title: Isoperimetry for two dimensional supercritical percolation

Abstract: Isoperimetric problems have been around since ancient history. They play an important role in many parts of mathematics as well as sciences in general. Isoperimetric inequalities and the shape of isoperimetric sets are generally well understood in Euclidean or other "nice" settings but are still subject of research in random domains, graphs, manifolds, etc. In my talk I will address the isoperimetric problem for one example of a random setting: the unique infinite connected component of supercritical bond percolation on the square lattice. In particular, I will sketch a proof of the fact that, as the volume of a (properly defined) isoperimetric set tends to infinity, its asymptotic shape can be characterized by an isoperimetric problem in the plane with respect to a particular (continuum) norm. As an application I will conclude that that the anchored isoperimetric profile with respect to a given point as well as the Cheeger constant of the giant component in finite boxes scale to deterministic quantities. This settles a conjecture of Itai Benjamini for the plane. Based on joint work with O. Louidor, E. Procaccia and R. Rosenthal.

• Friday, May 23rd: Arnab Sen - University of Minnesota.

Title: Continuous spectra for sparse random graphs

Abstract: The limiting spectral distributions of many sparse random graph models are known to contain atoms. But a more interesting question is when they also have some continuous part. In this talk, I will give affirmative answer to this question for several widely studied models of random graphs including Erdos-Renyi random graph G(n,c/n) with c > 1, random graphs with certain degree distributions and supercritical bond percolation on Z^2. I will also present several open problems. This is joint work with Charles Bordenave and Balint Virag.

• Thursday, May 29th ( Billingsley lecture @4:30 PM, Eckhart 133): Scott Sheffield - MIT.
• Friday, May 30th (regular probability seminar time and location): Scott Sheffield - MIT.

Title: Snowflakes, slot machines, Chinese dragons, and QLE

Abstract: What is the right way to think of a "random surface" or a "random planar graph"? How can one explain the dendritic patterns that appear in snowflakes, choral reefs, lightning bolts, and other physical systems, as well in as toy mathematical models inspired by these systems? How are these questions related to random walks and random fractal curves (in particular the famous SLE curves)? To begin to address these questions, I will introduce and explain the "quantum Loewner evolution", which is a family of growth processes closely related to SLE. I will explain. through pictures and animations and some discrete arguments, how QLE is defined and what role it might play in addressing the questions raised above. In a continuation of the talk on Friday afternoon (at the probability seminar), I will present a more analytic, continuum construction of QLE and discuss its relationship to the so-called Brownian map. Joint work with Jason Miller.

• Friday, June 6th: Laurence Field - University of Chicago.

Title: Two-sided radial SLE and length-biased chordal SLE

Abstract: Models in statistical physics often give measures on self-avoiding paths. We can restrict such a measure to the paths that pass through a marked point, obtaining a "pinned measure". The aggregate of the pinned measures over all possible marked points is just the original measure biased by the path's length. Does the analogous result hold for SLE curves, which appear in the scaling limits of many such models at criticality? We show that it does: the aggregate of two-sided radial SLE is length-biased chordal SLE, where the path's length is measured in the natural parametrization.

## Winter 2014 Seminars

• Friday, Jan 10th: Fredrik Vieklund - Columbia University / Uppsala University.

Title: Planar growth models and conformal mapping

Abstract: Random, fractal-like growth can be seen in several places in nature. Several mathematical models based in one way or another on harmonic measure exist, but despite significant efforts little is known about these models. I will survey some of the models and problems, focusing in particular on constructions based on conformal maps. Towards the end I will discuss some recent joint work with Sola and Turner on one of these models.

• Friday, Jan 24th: Double talks!
Edward Waymire - Oregon state university. (2:30-3:30)

Title: Tree Polymers Under Strong Disorder

Abstract: Tree polymers are simplifications of 1+1 dimensional lattice polymers made up of polygonal paths of a (nonrecombining) binary tree having random path probabilities. The path probabilities are (normalized) products of i.i.d. positive weights. As such, they reside in the more general framework of multiplicative cascades and branching random walk. The probability laws of these paths are of interest under weak and strong types of disorder. Some recent results, speculation and conjectures will be presented for this class of models under both weak and strong disorder conditions. This is based on various joint papers with Partha Dey, Torrey Johnson, or Stan Williams.

Wei Wu - Brown University. (3:35-4:35)

Title: Random fields from uniform spanning trees

Abstract: The uniform spanning tree (UST) is a fundamental combinatorial object. In two dimensions, using conformal invariance and planar duality, it is shown that the scaling limits of UST is given by one of the SLE path. We discuss the random field approach, and study the scaling limit of certain random fields coupled with USTs. This approach works on general graphs, and may help to understand the scaling limits of UST in higher dimensions. This talk is based on several joint works with Adrien Kassel, Richard Kenyon and Xin Sun.

• Friday, Jan 31th: Hao Wu - MIT.

Title: Intersections of SLE paths

Abstract: SLE curves are introduced by Oded Schramm as the candidate of the scaling limit of discrete models. In this talk, we first describe basic properties of SLE curves and their relation with discrete models. Then we summarize the Hausdorff dimension results related to SLE curves, in particular the new results about the dimension of cut points and double points. Third we introduce Imaginary Geometry, and from there give the idea of the proof of the dimension results.

• Friday, Feb 7th: Wei-Kuo Chen - Univ Chicago.

Title: On Gaussian inequalities for product of functions

Abstract: Gaussian inequalities have played important roles in various scientific areas. In this talk, we will present simple algebraic criteria that yield sharp Holder types of inequalities for the product of functions of Gaussian random vectors with arbitrary covariance structure. As an application, we will explain how our results yield several famous inequalities in functional geometry, such as, the Brascamp-Lieb inequality, the sharp Young inequality, etc. This part of the talk is based on the recent joint work with N. Dafnis and G. Paouris. Along this direction, we will discuss a conjecture on the convexity of the Parisi functional arising from the study of the Sherrington-Kirkpatrick model in spin glass.

• Friday, Feb 14th: Greg Lawler - Univ Chicago.

Title: Conformal invariance of the Green's function for loop-erased random walk

Abstract: The planar loop-erased random walk (LERW) is obtained from the usual random walk by erasing loops. The LERW is related to a number of other models such as the uniform spanning tree. We consider a fixed simply connected domain in C containing the origin, and two distinct boundary points a and b. For a fixed lattice spacing, we consider the probability that a LERW goes from a to b goes through an edge containing the origin. We show that the normalized limit of this probability goes to a conformally covariant quantity, the Green's function for the Schramm-Loewner evolution. This is joint work with Christian Benes and Fredrik Viklund .

• Friday, Feb 21st: Asaf Nachmias - UBC.

Title: Random walks on planar graphs via circle packings

Abstract: I will describe two results concerning random walks on planar graphs and the connections with Koebe's circle packing theorem (which I will not assume any knowledge of): 1. A bounded degree planar triangulation is recurrent if an only if the set of accumulation points of its circle packing is a polar set (that is, has zero logarithmic capacity). This extends a result of He and Schramm who proved recurrence (transience) when the set of accumulation points is empty (a closed Jordan curve). Joint work with Ori Gurel-Gurevich and Juan Souto. 2. The Poisson boundary (the space of bounded harmonic functions) of a transient bounded degree triangulation of the plane is characterized by the topological boundary obtained by circle packing the graph in the unit disk. In other words, any bounded harmonic function on the graph is the harmonic extension of some measurable function on the boundary of the unit disc. Joint work with Omer Angel, Martin Barlow and Ori Gurel-Gurevich.

• Friday, Feb 28th: Ronen Eldan - Microsoft Research, Redmond.

Title: A Two-Sided Estimate for the Gaussian Noise Stability Deficit

Abstract: The Gaussian Noise Stability of a set A in Euclidean space is the probability that for a Gaussian vector X conditioned to be in A, a small Gaussian perturbation of X will also be in A. Borell's celebrated inequality states that a half-space maximizes the noise stability among all possible sets having the same Gaussian measure. We present a novel short proof of this inequality, based on stochastic calculus. Moreover, we prove an almost tight, two-sided, dimension-free robustness estimate for this inequality: We show that the deficit between the noise stability of a set A and an equally probable half-space H can be controlled by a function of the distance between the corresponding centroids. As a consequence, we prove a conjecture of Mossel and Neeman, who used the total-variation distance as a metric.

• Friday, Mar 7th: Nikolaos Dafnis - Texas A&M University.

Title: Asymptotic behavior of log-concave probability measures

Abstract: A probability measure $\mu$ in ${\mathbb R}^n$ is called log-concave if $\mu\big(\lambda A + (1-\lambda) B\big) \geq \mu(A)^\lambda\,\mu(B)^{1-\lambda}$, for every $\lambda\in[0,1]$ and every $A,B$ Borel subsets of ${\mathbb R}^n$. Two basic examples are the uniform measure restricted to a convex body in ${\mathbb R}^n$ with volume $1$ (Brunn-Minkowski inequality) and the normal Gaussian measure in ${\mathbb R}^n$. We are studying the asymptotic behavior of some random geometric quantities such as the volume and the radius of a random polytope generated by sampling with respect to a log-concave probability measure. We will show that asymptotically ( as the dimension $n$ goes to infinity), they behave like if we had sampled with respect to the Gaussian measure.

## 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 beneﬁt 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 Gonçalves.

• 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.