## Probability and Statistical Physics Seminar

### University of Chicago

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

## Winter 2020 Seminars

• Friday, January 24: Patrick Lopatto - Harvard

Title: Spectral Statistics of Lévy Matrices

Abstract: Lévy matrices are symmetric random matrices whose entries are independent \alpha-stable laws. Such distributions have infinite variance, and when \alpha <1, infinite mean. In the latter case these matrices are conjectured to exhibit a sharp transition from a delocalized regime at low energy to a localized regime at high energy, like the infamous Anderson model in mathematical physics. We discuss work establishing the existence of a delocalized regime with GOE eigenvalue statistics. Further, we characterize the eigenvector statistics in this regime and find they display novel, non-Gaussian behavior. These describe joint works with Amol Aggarwal, Jake Marcinek, and Horng-Tzer Yau.

• Friday, January 31 (two talks)

Probability seminar: Joshua Pfeffer - MIT
2:30-3:30 PM, Eckhart 202

Title: Understanding Liouville quantum gravity through two square subdivision models

Abstract: In my talk I will discuss a general approach to better understand the geometry of Liouville quantum gravity (LQG). The idea, roughly speaking, is to partition the random surface into dyadic squares of roughly the same "LQG size''. Based on this approach, I will introduce two different models of LQG that will provide answers to three questions in the field:
1. Rigorously explain the so-called "DDK ansatz'' by proving that, for a surface with metric tensor some regularized version of the LQG heuristic metric tensor, its law corresponds to sampling a surface with probability proportional to the $(-c/2)$-th power of the zeta-regularized determinant of the Laplacian, with $c$ the matter central charge.
2. Provide a heuristic picture of the geometry of LQG with matter central charge in the interval $(1,25)$. (The geometry in this regime is mysterious even from a physics perspective.)
3. Explain why many works in the physics literature may have missed the nontrivial conformal geometry of LQG with matter central charge in the interval $(1,25)$ when they suggest (based on numerical simulations and heuristics) that LQG exhibits the macroscopic behavior of a continuum random tree in this phase.
This talk is based on a joint work with Morris Ang, Minjae Park, and Scott Sheffield; and a joint work with Ewain Gwynne, Nina Holden, and Guillaume Remy.

Statistics Colloquium: Yi Sun - Columbia
4:00-5:00 PM, Jones 303

Title: Fluctuations for Products of Random Matrices

Abstract: Products of large random matrices appear in many modern applications such ashigh dimensional statistics (MANOVA estimators), machine learning (Jacobians of neural networks), and population ecology (transition matrices of dynamical systems). Inspired by these situations, this talk concerns global limits and fluctuations of singular values of products of independent random matrices as both the size N and number M of matrices grow. As N grows, I will show for a variety of ensembles that fluctuations of the Lyapunov exponents converge to explicit Gaussian fields which transition from log-correlated for fixed M to having a white noise component for M growing with N. I will sketch our method, which uses multivariate generalizations of the Laplace transform based on the multivariate Bessel function from representation theory.

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

Title: Random walks in random environment on the strip

Abstract: The Random walks in random environment on the strip model was introduced by Bolthausen and Goldsheid in order to treat 1 dimensional Random walks in random environment with bounded jump. In this talk I will review recent results about this model.

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

Title: Community Detection in Sparse Random Hypergraphs

Abstract: The stochastic block model is a generative model for random graphs with a community structure, which has been one of the most fruitful research topics in community detection and clustering. A phase transition behavior for detection was conjecured by Decelle et al. (2011) at the Kesten-Stigum threshold, and was confirmed by Mossel et al. (2012, 2013) and Massoulié (2013). We consider the community detection problem in random hypergraphs. Angelini et al. (2015) conjectured a detection threshold in sparse hypergraphs generated by a hypergraph stochastic block model (HSBM). We confirmed the positive part of the phase transition by a generalization of the method developed in Massoulié (2013). We introduced a matrix which counts self-avoiding walks on hypergraphs, whose leading eigenvectors give us a correlated reconstruction of the community. In the course of proving our main result, we developed a moment method for sparse hypergraphs and constructed a coupling between the local neighborhood of HSBMs and multitype Poisson Galton-Watson hypertrees. This is joint work with Soumik Pal.

• Friday, March 13: Shirshendu Ganguly - University of California, Berkeley

Title: TBA

Abstract: TBA

## Spring 2020 Seminars

• Friday, April 3: Jack Hanson - City College of New York

Title: TBA

Abstract: TBA

• Friday, April 10: Gordon Slade - UBC

Title: TBA

Abstract: TBA

• Friday, April 17: Pratima Hebbar - Duke

Title: TBA

Abstract: TBA

• Billingsley speaker Alexei Borodin - MIT (two talks)

Thursday, April 23: Billingsley Lecture
Title: TBA
Abstract: TBA

Friday, April 24, 2:30 PM, E202: Probability Seminar
Title: TBA
Abstract: TBA

• Friday, May 1: Yilin Wang - MIT

Title: TBA

Abstract: TBA

• Friday, May 29: Linan Chen - McGill University

Title: TBA

Abstract: TBA

## Fall 2019 Seminars

• Friday, October 4: Vishesh Jain - MIT

Title: Title: A combinatorial approach to the quantitative invertibility of random matrices.

Abstract: Let $s_n(M_n)$ denote the smallest singular value of an $n\times n$ random matrix $M_n$. We will discuss a novel combinatorial approach (in particular, not using either inverse Littlewood--Offord theory or net arguments) for obtaining upper bounds on the probability that $s_n(M_n)$ is smaller than $\eta \geq 0$ for quite general random matrix models. Such estimates are a fundamental part of the non-asymptotic theory of random matrices and have applications to the strong circular law, numerical linear algebra etc. In several cases of interest, our approach provides stronger bounds than those obtained by Tao and Vu using inverse Littlewood--Offord theory.

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

• Friday, October 18: Oanh Nguyen - Princeton

Title: Roots of random functions

Abstract: The study of random functions has been investigated for many decades. In this talk, we will discuss several classical results by Kac, Littlewood-Offord, Erdos-Offord, etc. together with recent developments and open problems in the field. We also discuss a general framework to prove universality results for correlation functions of the roots and apply it to study various questions on random polynomials, random trigonometric functions, and random eigenfunctions. Using these universality results, we estimate the number of nodal intersections and the number of real roots. We also show that the number of real roots satisfies the Central Limit Theorem. This talk is based on several joint papers with Mei-Chu Chang, Yen Do, Hoi Nguyen, and Van Vu.

• Friday, October 25: Alisa Knizel - Columbia

Title: Asymptotics of discrete β-corners processes via discrete loop equations

Abstract: We introduce and study stochastic particle ensembles which are natural discretizations of general β-corners processes. We prove that under technical assumptions on a general analytic potential the global fluctuations for the difference between two adjacent levels are asymptotically Gaussian. The covariance is universal and remarkably differs from its counterpart in random matrix theory. Our main tools are certain novel algebraic identities that are multi-level analogues of the discrete loop equations. Based on joint work with Evgeni Dimitrov (Columbia University)

• Friday, November 1: Amir Dembo - Stanford

Title: Large deviations of subgraph counts for sparse random graphs.

Abstract: In this talk, based on joint works with Nick Cook and with Sohom Bhattacharya, I will discuss recent developments in the emerging theory of nonlinear large deviations, focusing on sharp upper tails for counts of several fixed subgraphs in a large sparse random graph, such as Erdos–Renyi or uniformly d-regular. Time permitting, I will describe our quantitative versions of the regularity and counting lemmas, which are geared for the study of sparse random graphs in the large deviations regime, and what our results suggest regarding certain questions in extremal graph theory.

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

Title: Distorted Brownian motion on spaces with varying dimension

Abstract: We introduce "distorted Brownian motion" (dBM) on a state space with varying dimension. Roughly speaking, the state space consists of two components: a 3-dimensional component and a 1-dimensional component. These two parts are joined together at the origin. The restriction of dBM on the 3-d component models a homopolymer with attractive potential at the origin. The restriction of dBM on the 1-d component also receives a push towards the origin. Such a process can be nicely characterized in terms of Dirichlet form, and we can find its density estimate by characterizing its radial process.

• Friday, November 15: no seminar

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

Title: Fermionic eigenvector moment flow

Abstract: We first present known results and open problems on the study of eigenvector statistics of large random matrices such as complete delocalization, quantum unique ergodicity or asymptotic entry distribution. We willl see how we can obtain some of these properties dynamically using the Dyson Brownian motion and the Bourgade-Yau eigenvector moment flow: it consists of a parabolic equation followed by eigenvector moments. We will then present new moment observables which follow a similar equation and which can be seen as a Fermionic counterpart to the (Bosonic) original ones. By combining the information obtained through the study of these two families of observables, we can compute, previously intractable, correlations between eigenvectors.

• Friday, November 29: no seminar (Thanksgiving)

• Friday, December 6: Ewain Gwynne - Cambridge

## Fall 2013 Seminars

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

Title: Performance of the Metropolis algorithm on a disordered tree: the Einstein relation.

Abstract: Consider a d-ary rooted tree (d>2) where each edge e is assigned an i.i.d. (bounded) random variable X(e) of negative mean. Assign to each vertex v the sum S(v) of X(e) over all edges connecting v to the root, and assume that the maximum S_n* of S(v) over all vertices v at distance n from the root tends to infinity (necessarily, linearly) as n tends to infinity. We analyze the Metropolis algorithm on the tree and show that under these assumptions there always exists a temperature of the algorithm so that it achieves a linear (positive) growth rate in linear time. This confirms a conjecture of Aldous (Algorithmica, 22(4):388-412, 1998). The proof is obtained by establishing an Einstein relation for the Metropolis algorithm on the tree. Joint work with Pascal Maillard.

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

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

Title: Persistence Probabilities.

Abstract: Persistence probabilities concern how likely it is that a stochastic process has a long excursion above fixed level and of what are the relevant scenarios for this behavior. Power law decay is expected in many cases of physical significance and the issue is to determine its power exponent parameter. I will survey recent progress in this direction (jointly with Sumit Mukherjee), dealing with stationary Gaussian processes that arise from random algebraic polynomials of independent coefficients and from the solution to heat equation initiated by white noise. If time permits, I will also discuss the relation to joint works with Jian Ding and Fuchang Gao, about persistence for iterated partial sums and other auto-regressive sequences, and to the work of Sakagawa on persistence probabilities for the height of certain dynamical random interface models.

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

Title: Statistics on Hilbert's Sixteenth Problem

Abstract: The first part of Hilbert's sixteenth problem concerns real algebraic geometry: We are asked to study the number and possible arrangements of the connected components of a real algebraic curve (or hypersurface). I will describe a probabilistic approach to studying the topology, volume, and arrangement of the zero set (in real projective space) of a random homogeneous polynomial. The outcome depends on the definition of "random". A popular Gaussian ensemble uses monomials as a basis, but we will favor eigenfunctions on the sphere (spherical harmonics) as a basis. As we will see, this "random wave" model produces a high expected number of components (a fraction of the Harnack bound that was an inspiration for Hilbert's sixteenth problem). This is joint work with Antonio Lerario.

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

Title: A Dynamic Graph Model of Barter Exchanges

Abstract: Motivated by barter exchanges, we study average waiting time in a dynamic random graph model. A node arrives at each time step. A directed edge is formed independently with probability p with each node currently in the system. If a cycle is formed, of length no more than 3, then that cycle of nodes is removed immediately. We show that the average waiting time for > a node scales as 1/p^{3/2} for small p, for this policy. Moreover, we prove that we cannot achieve better delay scaling by batching. Our results through new light on the operation of kidney exchange programs. The insight offered by our analysis is that the beneÃ¯Â¬Ât of waiting for additional incompatible patient-donor pairs to arrive (batching) into kidney exchange clearinghouses is not substantial and is outweighed by the cost of waiting. Joint work with Ross Anderson, Itai Ashlagi and David Gamarnik.

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

Title: Epsilon-biased sets, the Legendre symbol, and getting by with a few random bits

Abstract: Subsets of F_2^n that are p-biased, meaning that the parity of any set of bits is even or odd with probability close to 1/2, are useful tools in derandomization. They also correspond to optimal error-correcting codes,i.e. meeting the Gilbert-Varshamov bound, with distance close to n/2. A simple randomized construction shows that such sets exist of size O(n/p^2); recently, Ben-Aroya and Ta-Shma gave a deterministic construction of size O((n/p^2)^(5/4)). I will review deterministic constructions of Alon, Goldreich, Haastad, and Peralta of sets of size O(n/p^3) and O(n^2/p^2), and discuss the delightful pseudorandom properties of the Legendre symbol along the way. Then, rather than derandomizing these sets completely in exchange for making them larger, we will try moving in a different direction on the size-randomness plane, constructing sets of optimal size O(n/p^2) with as few random bits as possible. The naive randomized construction requires O(n^2/p^2) random bits. I will show that this can be reduced to O(n log(n/p)) random bits. Like Alon et al., our construction uses the Legendre symbol and Weil sums, but in a different way to control high moments of the bias. I'll end by saying a few words about Ramsey graphs and random polynomials. This is joint work with Alex Russell.

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

Title: Quantum spin systems and graphical representations

Abstract: Quantum spin systems are mathematical models for magnetism. But the quantum nature is a difficulty. For some models there are graphical representations, which relate to interacting particle processes (with some changes). I will discuss one application done jointly with Nick Crawford and Stephen Ng, called emptiness formation probability where this approach works.

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

Title: Delocalization of eigenvectors of random matrices

Abstract: Eigenvectors of random matrices are much less studied than eigenvalues, despite their importance. The simplest question is whether the eigenvectors are delocalized, i.e. all of their coordinates are as small as can be, of order n^{-1/2}. Even this simple looking problem has been open until very recently. Currently there are two approaches to delocalization - spectral (via local eigenvalue statistics) and geometric (via high dimensional probability). This talk will explain these approaches and popularize related open problems. Based on joint work with Mark Rudelson (Michigan).

• Friday, Nov 29th: Thanksgiving

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

Title: Multiple Phase Transitions for long range first-passage percolation on lattices

Abstract: Given a graph G with non-negative edge weights, the passage time of a path is the sum of weights of the edges in the path, and the first-passage time to reach u from v is the minimum passage time of a path joining them. We consider a long range first-passage model on Z^d in which, the weight w(x,y) of the edge joining x and y has exponential distribution with mean |x-y|^a for some fixed a > 0, and the edge weights are independent. We analyze the growth of the set of vertices reachable from the origin within time t, and show that there are four different growth regimes depending on the value of a. Joint work with Partha Dey.

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

Title: Statistics on Hilbert's Sixteenth Problem

Abstract: The first part of Hilbert's sixteenth problem concerns real algebraic geometry: We are asked to study the number and possible arrangements of the connected components of a real algebraic curve (or hypersurface). I will describe a probabilistic approach to studying the topology, volume, and arrangement of the zero set (in real projective space) of a random homogeneous polynomial. The outcome depends on the definition of "random". A popular Gaussian ensemble uses monomials as a basis, but we will favor eigenfunctions on the sphere (spherical harmonics) as a basis. As we will see, this "random wave" model produces a high expected number of components (a fraction of the Harnack bound that was an inspiration for Hilbert's sixteenth problem). This is joint work with Antonio Lerario.

## Spring 2013 Seminars

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

Title: On the monotonicity of the speed of biaised random walk on a Galton-Watson tree without leaves.

Abstract: We will present different results related to the speed of biased random walks in random environments. Our focus will be on a recent paper by Ben Arous, Fribergh and Sidoravicius proving that the speed of the biased random walk on a Galton-Watson tree without leaves is increasing for high biases. This partially solves a question asked by Lyons, Pemantle and Peres.

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

Title: Search Games, The Cauchy process and Optimal Kakeya Sets

Abstract: A planar set that contains a unit segment in every direction is called a Kakeya set. These sets have been studied intensively in geometric measure theory and harmonic analysis since the work of Besicovich (1928); we find a new connection to game theory and probability via a search game first analyzed by Adler et al (2003). A hunter and a rabbit move on the n-vertex cycle without seeing each other. At each step, the hunter moves to a neighboring vertex or stays in place, while the rabbit is free to jump to any node. Thus they are engaged in a zero sum game, where the payoff is the capture time. The known optimal randomized strategies for hunter and rabbit achieve expected capture time of order n log n. We show that every rabbit strategy yields a Kakeya set; the optimal rabbit strategy is based on a discretized Cauchy random walk, and it yields a Kakeya set K consisting of 4n triangles, that has minimal area among such sets (the area of K is of order 1/log(n)). Passing to the scaling limit yields a simple construction of a random Kakeya set with zero area from two Brownian motions. (Joint work with Y. Babichenko, R. Peretz, P. Sousi and P. Winkler).

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

Tutorial Seminar: What is the mixing time for random walk on a graph?

Abstract: Consider a simple random walk on a finite graph. The mixing time is the time it takes the walk to reach a position that is approximately independent of the starting point; it has been studied intensively by combinatorialists, computer scientists and probabilists; the mixing time arises in statistical physics as well. Applications of mixing times range from random sampling and card shuffling, to understanding convergence to equilibrium in the Ising model. It is closely related to expansion and eigenvalues. Besides introducing this topic, I will also describe the open problem of understanding which random walks exhibit "cutoff", a sharp transition to stationarity first discovered by Diaconis, Shashahani and Aldous in the early 1980s but still mysterious.

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

Title: Convergence to equilibrium for nonreversible diffusions.

Abstract: The problem of convergence to equilibrium for diffusion processes is of theoretical as well as applied interest, for example in nonequilibrium statistical mechanics and in statistics, in particular in the study of Markov Chain Monte Carlo (MCMC) algorithms. Powerful techniques from analysis and PDEs, such as spectral theory and functional inequalities (e.g. logarithmic Sobolev inequalities) can be used in order to study convergence to equilibrium. Quite often, the diffusion processes that appear in applications are degenerate (in the sense that noise acts directly to only some of the degrees of freedom of the system) and/or nonreversible. The study of convergence to equilibrium for such systems requires the study of non-selfadjoint, possibly non-uniformly elliptic, second order differential operators. In this talk we show how the recently developed theory of hypocoercivity can be used to prove exponentially fast convergence to equilibrium for such diffusion processes. Furthermore, we will show how the addition of a nonreversible perturbation to a reversible diffusion can speed up convergence to equilibrium. This is joint work with M. Ottobre, K. Pravda-Starov, T. Lelievre and F. Nier.

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

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

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

Title: Random Walk with Reinforcement

Abstract: Picture a triangle, with vertices labeled A, B, C. A random walker starts at A and chooses a random nearest neighbor. At each stage, the walker adds 1 to the weight of each crossed edge and chooses the next step with probability proportional to the current edge weights. The question is 'what happens?'. This simple problem leads into interesting corners: to Bayesian analysis of the transition mechanism of Markov chains (and protein folding) and to the hyperbolic sigma model of statistical physics. Work of (and with) Billingsley, Baccalado, Freedman, Tarres, and Sabot will be reviewed.

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

Title: Exchangeable random measures

Abstract: Classical theorems of de Finetti, Aldous-Hoover and Kallenberg describe the structure of exchangeable probability measures on spaces of sequences or arrays. Similarly, one can add an extra layer of randomness, and ask after exchangeable random measures on these spaces. It turns out that those classical theorems, coupled with an abstract version of the `replica trick' from statistical physics, give a structure theorem for these random measures also. This leads to a new proof of the Dovbysh-Sudakov Theorem describing exchangeable positive semi-definite matrices.

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

Title: Maximum independent sets in random d-regular graphs

Abstract: Satisfaction and optimization problems subject to random constraints are a well-studied area in the theory of computation. These problems also arise naturally in combinatorics, in the study of sparse random graphs. While the values of limiting thresholds have been conjectured for many such models, few have been rigorously established. In this context we study the size of maximum independent sets in random d-regular graphs. We show that for d exceeding a constant d(0), there exist explicit constants A, C depending on d such that the maximum size has constant fluctuations around A*n-C*(log n) establishing the one-step replica symmetry breaking heuristics developed by statistical physicists. As an application of our method we also prove an explicit satisfiability threshold in random regular k-NAE-SAT. This is joint work with Jian Ding and Allan Sly.

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

Title: Scaling limit of the abelian sandpile

Abstract: Which functions of two real variables can be expressed as limits of superharmonic functions from (1/n)Z2 to (1/n2)Z? I'll discuss joint work with Wesley Pegden and Charles Smart on the case of quadratic functions, where this question has a surprising and beautiful answer: the maximal such quadratics are classified by the circles in a certain Apollonian circle packing of the plane. I'll also explain where the question came from (the title is a hint!).

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

Title: The relaxation of a family of broken bond crystal surface models

Abstract: We study the continuum limit of a family of kinetic Monte Carlo models of crystal surface relaxation that includes both the solid-on-solid and discrete Gaussian models. With computational experiments and theoretical arguments we are able to derive several partial differential equation (PDE) limits identified (or nearly identified) in previous studies and to clarify the correct choice of surface tension appearing in the PDE and the correct scaling regime giving rise to each PDE. We also provide preliminary computational investigations of a number of interesting qualitative features of the large scale behavior of the models.

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

Title: Random polymers and last passage percolation: variational formulas, Busemann functions, geodesics, and other stories

Abstract: We give variational formulas for random polymer models, both in the positive- and zero-temperature cases. We solve these formulas in the oriented two-dimensional zero-temperature case. The solution comes via proving almost-sure existence of the so-called Busemann functions. We then use these results to prove existence, uniqueness, and coalescence of semi-infinite directional geodesics, for exposed points of differentiability of the limiting shape function.

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

Title: The scaling limit of simple triangulations and quadrangulations

Abstract: A graph is simple if it contains no loops or multiple edges. We establish Gromov--Hausdorff convergence of large uniformly random simple triangulations and quadrangulations to the Brownian map, answering a question of Le Gall (2011). In proving the preceding fact, we introduce a labelling function for the vertices of the triangulation. Under this labelling, distances to a distinguished point are essentially given by vertex labels, with an error given by the winding number of an associated closed loop in the map. The appearance of a winding number suggests that a discrete complex-analytic approach to the study of random triangulations may lead to further discoveries. Joint work with Marie Albenque.

## Winter 2013 Seminars

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

Title Law of the extremes for the two-dimensional discrete Gaussian Free Field

Abstract: A two-dimensional discrete Gaussian Free Field (DGFF) is a centered Gaussian process over a finite subset (say, a square) of the square lattice with covariance given by the Green function of the simple random walk killed upon exit from this set. Recently, much effort has gone to the study of the concentration properties and tail estimates for the maximum of DGFF. In my talk I will address the limiting extreme-order statistics of DGFF as the square-size tends to infinity. In particular, I will show that for any sequence of squares along which the centered maximum converges in law, the (centered) extreme process converges in law to a randomly-shifted Gumbel Poisson point process which is decorated, independently around each point, by a random collection of auxiliary points. If there is any time left, I will review what we know and/or believe about the law of the random shift. This talk is based on joint work with Oren Louidor (UCLA).

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

Title: The Virasoro algebra and discrete Gaussian free field

Abstract: The Virasoro algebra is an infinite dimensional Lie algebra that plays an important role in the Conformal Field Theory (CFT) methods employed by physicists to describe and study conformally invariant scaling limits of planar critical lattice models from statistical physics. Despite much progress in the last decade, it seems fair to say that from a mathematical perspective many aspects of the connections between discrete model and continuum limit CFT remain somewhat mysterious. In the talk I will discuss recent joint work with C. Hongler and K. Kytola concerning the discrete Gaussian free field on a square grid. I will explain how for this model discrete complex analysis can be used to construct explicit (exact) representations of the Virasoro algebra of central charge 1 directly on the discrete level.

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

Title: Markov type and the multi-scale geometry of metric spaces

Abstract: The behavior of random walks on metric spaces can sometimes be understood by embedding such a walk into a nicer space (e.g. a Hilbert space) where the geometry is more readily approachable. This beautiful theme has seen a number of geometric and probabilistic applications. We offer a new twist on this study by showing that one can employ mappings that are significantly weaker than bi-Lipschitz. This is used to answer questions of Naor, Peres, Schramm, and Sheffield (2004) by proving that planar graph metrics and doubling metrics have Markov type 2. The main new technical idea is that martingales are significantly worse at aiming than one might at first expect. Joint work with Jian Ding and Yuval Peres.

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

Title: The Renormalization Group for Disordered Systems

Abstract: We investigate the Renormalization Group (RG) approach in finite- dimensional glassy systems, whose critical features are still not well-established, or simply unknown. We focus on spin and structural-glass models built on hierarchical lattices, which are the simplest non-mean-field systems where the RG framework emerges in a natural way. The resulting critical properties shed light on the critical behavior of spin and structural glasses beyond mean field, and suggest future directions for understanding the criticality of more realistic glassy systems.

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

Title: Geodesics and Direction in 2d First-Passage Percolation

Abstract: I will discuss geodesics in first-passage percolation, a model for fluid flow in a random medium. There are numerous conjectures about the existence, coalescence, and asymptotic direction of infinite geodesics under the model's random metric. C. Newman and collaborators have proved some of these under strong assumptions. I will explain recent results with Michael Damron which develop a framework for addressing these questions; this framework allows us to prove versions of Newman's results under minimal assumptions.

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

Title: Gaussian Free Field fluctuations for general-beta random matrix ensembles.

Abstract: It is now known that the asymptotic fluctuations of the height function of uniformly random lozenge tilings of planar domains (equivalently, stepped surfaces in 3d space) are governed by the Gaussian Free Field (GFF), which is a 2d analogue of the Brownian motion. On the other hand, in certain limit regimes such tilings converge to various random matrix ensembles corresponding to beta=2. This makes one wonder whether GFF should also somehow arise in general-beta random matrix ensembles. I will explain that this is indeed true and the asymptotics of fluctuations of classical general-beta random matrix ensembles is governed by GFF. This is joint work with A.Borodin.

• Friday, Mar 8th: No seminar.

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

Title: About heavy tailed random matrices.

Abstract:We investigate the behaviour of matrices which do not belong to the universality class of Wigner matrices because their entries have heavy tails.

## Fall 2012 Seminars

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

Title: Chaos problem in mean field spin glasses

Abstract: The main objective in spin glasses from the physical perspective is to understand the strange magnetic properties of certain alloys. Yet the models invented to explain the observed phenomena are also of a rather fundamental nature in mathematics. In this talk we will first introduce the famous Sherrington-Kirkpatrick model as well as some known results about this model such as the Parisi formula and the limiting behavior of the Gibbs measure. Next, we will discuss the problems of chaos in the mixed p-spin models and present mathematically rigorous results including disorder, external field, and temperature chaos.

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

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

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

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

Title: The 2D SOS Model

Abstract: We present new results on the (2+1)-dimensional Solid-On-Solid model at low temperatures. Bricmont, El-Mellouki and Froelich (1986) showed that in the presence of a floor there is an entropic repulsion phenomenon, lifting the surface to a height which is logarithmic in the side of the box. We refine this and establish the typical height of the SOS surface is precisely the floor of [1/(4\beta)\log n], where n is the side-length of the box and \beta is the inverse-temperature. We determine the asymptotic shape of the top plateau and show that its boundary fluctuation are n^{1/3+o(1)}. Based on joint works with Pietro Caputo, Eyal Lubetzky, Fabio Martinelli and Fabio Toninelli.

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

Title: Spiking the random matrix hard edge.

Abstract: The largest eigenvalue of a finite rank perturbation of a random hermitian matrix is known to exhibit a phase transition (in the infinite dimensional limit). If the perturbation is small one sees the famous Tracy-Widom law, while a large perturbation results in a Gaussian fluctuation. In between there exists is a scaling window about a critical perturbation value leading to a separate family of limit laws. This basic discovery is due to Baik, Ben Arous, and Peche. More recently Bloemendal and Virag have shown this picture persists in the context of the general beta ensembles, giving new formulations of the critical limit laws . Yet another route, explained here, is to go through the random matrix hard edge, perturbing the smallest eigenvalues in the sample covariance set-up. A limiting procedure then recovers all the alluded to distributions. (Joint work with Jose Ramirez.)

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

Title: Directed polymers and the stochastic heat equation

Abstract: We show how some properties of the solutions of the Stochastic Heat Equation (SHE) can be derived from directed polymers in random environment. In particular, we show: * A new proof of the positivity of the solutions of the SHE * Improved bounds on the negative moments of the SHE * Results on the fluctuations of the log of the SHE in equilibrium, namely, the Cole-Hopf solution of the KPZ equation (if time allows).

• Friday, Nov 9th: Milton Jara - IMPA

Title: Second-order Boltzmann-Gibbs principle and applications

Abstract: The celebrated Botzmann-Gibbs principle introduced by Rost in the 80's roughly says the following. For stochastic systems with one or more conservation laws, fluctuations of the non-conserved quantities are faster than fluctuations of the conserved quantities. Therefore, in the right space-time window, the space-time fluctuations of a given observable are asymptotically equivalent to a linear functional of the conserved quantities. In one dimension, we prove two generalizations of this principle: a non-linear (or second-order) and a local version of it. This result opens a way to show convergence of fluctuations for non-linear models, like the ones on the fashionable KPZ universality class. As a corollary, we prove new convergence results for various observables of the asymmetric exclusion process, given in terms of solutions of the KPZ equation. Joint work with Patricia GonÃƒÂ§alves.

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

Title: SLE curves and natural parametrization

• Friday, Nov 23rd: Thanksgiving.

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

Title: Robust Gaussian noise stability

Abstract: Given two Gaussian vectors that are positively correlated, what is the probability that they both land in some fixed set A? Borell proved that this probability is maximized (over sets A with a given volume) when A is a half-space. We will give a new and simple proof of this fact, which also gives some stronger results. In particular, we can show that half-spaces uniquely maximize the probability above, and that sets which almost maximize this probability must be close to half-spaces.

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

Title: Spiking the random matrix hard edge.

Abstract: The largest eigenvalue of a finite rank perturbation of a random hermitian matrix is known to exhibit a phase transition (in the infinite dimensional limit). If the perturbation is small one sees the famous Tracy-Widom law, while a large perturbation results in a Gaussian fluctuation. In between there exists is a scaling window about a critical perturbation value leading to a separate family of limit laws. This basic discovery is due to Baik, Ben Arous, and Peche. More recently Bloemendal and Virag have shown this picture persists in the context of the general beta ensembles, giving new formulations of the critical limit laws . Yet another route, explained here, is to go through the random matrix hard edge, perturbing the smallest eigenvalues in the sample covariance set-up. A limiting procedure then recovers all the alluded to distributions. (Joint work with Jose Ramirez.)

## Winter/Spring 2012 Seminars

• Friday, Jan 20: Jian Ding - Stanford University

Title: Extreme values for random processes of tree structures

Abstract: The main theme of this talk is that studying implicit tree structures of random processes is of significance in understanding their extreme values. I will illustrate this by several examples including cover times for random walks, maxima for two-dimensional discrete Gaussian free fields, and stochastic distance models. Our main results include (1) An approximation of the cover time on any graph up to a multiplicative constant by the maximum of the Gaussian free field, which yields a deterministic polynomial-time approximation algorithm for the cover time (D.-Lee-Peres 2010); the asymptotics for the cover time on a bounded-degree graph by the maximum of the GFF (D. 2011); a bound on the cover time fluctuations on the 2D lattice (D. 2011). (2) Exponential and doubly exponential tails for the maximum of the 2D GFF (D. 2011); some results on the extreme process of the 2D GFF (D.-Zeitouni, in preparation). (3) Critical and near-critical behavior for the mean-field stochastic distance model (D. 2011).

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

Title: Imaginary Geometry and the Gaussian Free Field

Abstract: The Schramm-Loewner evolution (SLE) is the canonical model of a non-crossing conformally invariant random curve, introduced by Oded Schramm in 1999 as a candidate for the scaling limit of loop erased random walk and the interfaces in critical percolation. The development of SLE has been one of the most exciting areas in probability theory over the last decade because Schramm's curves have now been shown to arise as the scaling limit of the interfaces of a number of different discrete models from statistical physics. In this talk, I will describe how SLE curves can be realized as the flow lines of a random vector field generated by the Gaussian free field, the two-time-dimensional analog of Brownian motion. I will also explain how this perspective can be used to prove several new results regarding the sample path behavior of SLE, in particular reversibility for kappa in (4,8). Based on joint works with Scott Sheffied.

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

Title: Directed random polymers and Macdonald processes

Abstract: The goal of the talk is to survey recent progress in understanding statistics of certain exactly solvable growth models, particle systems, directed polymers in one space dimension, and stochastic PDEs. A remarkable connection to representation theory and integrable systems is at the heart of Macdonald processes, which provide an overarching theory for this solvability. This is based off of joint work with Alexei Borodin.

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

Title: Path properties of the Schramm-Loewner Evolution.

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

Title: Extrema of branching Brownian motion

Abstract: Branching Brownian motion (BBM) on the real line is a particle system where particles perform Brownian motion and independently split into two independent Brownian particles after an exponential holding time. The statistics of extremal particles of BBM in the limit of large time are of interest for physicists and probabilists since BBM constitutes a borderline case, among Gaussian processes, where correlations affect the statistics. In this talk, I will start by reviewing results on the law of the maximum of BBM (the rightmost particle), and present new results on the joint distribution of particles close to the maximum. In particular, I will show how the approach can be used to prove ergodicity of the particle system. If time permits, I will explain how the program for BBM lays out a road map to understand extrema of log-correlated Gaussian fields such as the 2D Gaussian free field. This is joint work with A. Bovier and N. Kistler.

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

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

Title: Large Deviations with Applications to Random Matrices and Random Graphs

Abstract: See it here.

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

Title: Large Deviations for an Unusual Sum

Abstract: See it here.

## Fall 2011 Seminars

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

Title: Landscape of random functions in many dimensions via Random Matrix Theory.

Abstract: How many critical values a typical Morse function have on a high dimensional manifold? Could we say anything about the topology of its level sets? In this talk I will survey a joint work with Gerard Ben Arous and Jiri Cerny that addresses these questions in a particular but fundamental example. We investigate the landscape of a general Gaussian random smooth function on the N-dimensional sphere. These corresponds to Hamiltonians of well-known models of statistical physics, i.e spherical spin glasses. Using the classical Kac-Rice formula, this counting boils down to a problem in Random Matrix Theory. This allows us to show an interesting picture for the complexity of these random Hamiltonians, for the bottom of the energy landscape, and in particular a strong correlation between the index and the critical value. We also propose a new invariant for the possible transition between the so-called 1-step replica symmetry breaking and a Full Replica symmetry breaking scheme and show how the complexity function is related to the Parisi functional.

• Friday, Oct 7: Antti Knowles - Harvard University

Title: Finite-rank deformations of Wigner matrices.

Abstract: The spectral statistics of large Wigner matrices are by now well-understood. They exhibit the striking phenomenon of universality: under very general assumptions on the matrix entries, the limiting spectral statistics coincide with those of a Gaussian matrix ensemble. I shall talk about Wigner matrices that have been perturbed by a finite-rank matrix. By Weyl's interlacing inequalities, this perturbation does not affect the large-scale statistics of the spectrum. However, it may affect eigenvalues near the spectral edge, causing them to break free from the bulk spectrum. In a series of seminal papers, Baik, Ben Arous, and Peche (2005) and Peche (2006) established a sharp phase transition in the statistics of the extremal eigenvalues of perturbed Gaussian matrices. At the BBP transition, an eigenvalue detaches itself from the bulk and becomes an outlier. I shall report on recent joint work with Jun Yin. We consider an NxN Wigner matrix H perturbed by an arbitrary deterministic finite-rank matrix A. We allow the eigenvalues of A to depend on N. Under optimal (up to factors of log N) conditions on the eigenvalues of A, we identify the limiting distribution of the outliers. We also prove that the remaining eigenvalues "stick" to eigenvalues of H, thus establishing the edge universality of H + A. On the other hand, our results show that the distribution of the outliers is not universal, but depends on the distribution of H and on the geometry of the eigenvectors of A. As the outliers approach the bulk spectrum, this dependence is washed out and the distribution of the outliers becomes universal.

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

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

Jonathan Mattingly - Duke University

Title: A Menagerie of Stochastic Stabilization

Abstract: A basic problem for a stochastic system is to show that it possesses a unique steady state which dictates the long term statistics of the system. Sometimes the existence of such a measure is the difficult part. One needs control of the excursions away from the systems typical scale. As in deterministic system, one popular method is the construction of a Lyapunov Function. In the stochastic setting there lack of systematic methods to construct a Lyapunov Function when the interplay between the deterministic dynamics and stochastic dynamics are important for stabilization. I will give some modest steps in this direction which apply to a number of cases. In particular I will show a system where an explosive deterministic system is stabilized by the addition of noise and examples of physical systems where it is not clear how the deterministic system absorbs the stochastic excitation with out blowing up.

• Friday, Oct 21: Vladas Sidoravicius - IMPA

Title: From random interlacements to coordinate and infinite cylinder percolation

Abstract: During the talk I will focus on the connectivity properties of three models with long (infinite) range dependencies: Random Interlacements, percolation of the vacant set in infinite rod model and Coordinate percolation. The latter model have polynomial decay in sub-critical and super-critical regime in dimension 3. I will explain the nature of this phenomenon and why it is difficult to handle these models technically. In the second half of the talk I will present key ideas of the multi-scale analysis which allows to reach some conclusions. At the end I will discuss applications and several open problems.

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

Title: Complete matchings and random matrix theory

Abstract: Over the last decade or so, it has been found that the distributions that first appeared in random matrix theory describe several objects in probability and combinatorics which do not come from matrix at all. We consider one such example from the so-called maximal crossing and nesting of random complete matchings of integers. We also discuss related non-intersecting process. This is a joint work with Bob Jenkins.

• Friday, Nov 11: Michael Damron - Princeton University

Title: A simplified proof of the relation between scaling exponents in first-passage percolation

Abstract: In first passage percolation, we place i.i.d. non-negative weights on the nearest-neighbor edges of Z^d and study the induced random metric. A long-standing conjecture gives a relation between two "scaling exponents": one describes the variance of the distance between two points and the other describes the transversal fluctuations of optimizing paths between the same points. This is sometimes referred to as the "KPZ relation." In a recent breakthrough work, Sourav Chatterjee proved this conjecture using a strong definition of the exponents. I will discuss work I just completed with Tuca Auffinger, in which we introduce a new and intuitive idea that replaces Chatterjee's main argument and gives an alternative proof of the relation. One advantage of our argument is that it does not require a certain non-trivial technical assumption of Chatterjee on the weight distribution.

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

Ofer Zeitouni - University of Minnessota

Title: Traveling waves, branching random walks, and the Gaussian free field

Abstract: I will discuss several aspects of Branching random walks and their relation with the KPP equation on the one hand, and the maximum of certain (two dimensional) Gaussian fields on the other. I will not assume any knowledge about either of these terms.

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

Title: The parafermionic observable in Schramm-Loewner Evolutions

Abstract: In recent years, work by Stanislav Smirnov and his co-authors has greatly advanced our understanding of discrete stochastic processes, such as self-avoiding walk and the Ising model, via the use of a tool known as the parafermionic observable. Much of that work has been done in order to show convergence of these models to Schramm-Loewner Evolutions (SLE) in the scaling limit, although very little work has been done on what the parafermionic observable is in SLE itself. In this talk I will introduce the parafermionic observable, and then discuss one possible generalization to the continuous setting. I will then briefly introduce SLE and compute its parafermionic observable, ending with a couple of open questions.

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

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

Title: The contact process on the complete graph with random, vertex-dependent infection rates.

Abstract: The contact process is an interacting particle system that is a very simple model for the spread of an infection or disease on a network. Traditionally, the contact process was studied on homogeneous graphs such as the integer lattice or regular trees. However, due to the non-homogeneous structure of many real-world networks, there is currently interest in studying interacting particle systems in non-homogeneous graphs and environments. In this talk, I consider the contact process on the complete graph, where the vertices are assigned (random) weights and the infection rate between two vertices is proportional to the product of their weights. This set-up allows for some interesting analysis of the process and detailed calculations of phase transitions and critical exponents.

• Friday, Dec 9: Paul Bougarde - Harvard University

Title: Universality for beta-ensembles.

Abstract: Wigner stated the general hypothesis that the distribution of eigenvalue spacings of large complicated quantum systems is universal in the sense that it depends only on the symmetry class of the physical system but not on other detailed structures. The simplest case for this hypothesis is for ensembles of large but finite dimensional matrices. Spectacular progress was done in the past decade to prove universality of random matrices presenting an orthogonal, unitary or symplectic invariance. These models correspond to log-gases with respective inverse temperature 1, 2 or 4. I will report on a joint work with L. Erd\"os and H.-T. Yau, which yields universality for the log-gases at arbitrary temperature. The involved techniques include a multiscale analysis and a local logarithmic Sobolev inequality.

### Past Seminars

Friday, Oct. 8, Fredrik Johansson Viklund, Columbia U.

Friday, Oct. 15, Midwest Probability Colloquium at Northwestern

Friday, Oct. 29, Tom Alberts, U. of Toronto, Convergence of Loop-Erased Random Walk to SLE(2) in the Natural Time Parameterization

I will discuss work in progress with Michael Kozdron and Robert Masson on the convergence of the two-dimensional loop-erased random walk process to SLE(2), with the time parameterization of the curves taken into account. This is a strengthening of the original Lawler, Schramm, and Werner result which was only for curves modulo a reparameterization. The ultimate goal is to show that the limiting curve is SLE(2) with the very specific natural time parameterization that was recently introduced in Lawler and Sheffield, and further studied in Lawler and Zhou. I will describe several possible choices for the parameterization of the discrete curve that should all give the natural time parameterization in the limit, but with the key difference being that some of these discrete time parameterizations are easier to analyze than the others.

Friday, Dec. 3, Pierre Nolin, Courant Institute Connection probabilities and RSW-type bounds for the two-dimensional FK Ising model

For two-dimensional independent percolation, Russo-Seymour-Welsh (RSW) bounds on crossing probabilities are an important a-priori indication of scale invariance, and they turned out to be a key tool to describe the phase transition: what happens at and near criticality. In this talk, we prove RSW-type uniform bounds on crossing probabilities for the FK Ising model at criticality, independent of the boundary conditions. A central tool in our proof is Smirnov's fermionic observable for the FK Ising model, that makes some harmonicity appear on the discrete level, providing precise estimates on boundary connection probabilities. We also prove several related results - including some new ones - among which the fact that there is no magnetization at criticality, tightness properties for the interfaces, and the value of the half-plane one-arm exponent. This is joint work with H. Duminil-Copin and C. Hongler.