Thursdays in 901 Van Vleck Hall at 2:25 PM, unless otherwise noted. We usually end for questions at 3:15 PM.
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Friday, August 10, 10am, B239 Van Vleck András Mészáros, Central European University, Budapest
Title: The distribution of sandpile groups of random regular graphs
Abstract: We study the distribution of the sandpile group of random -regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the -Sylow subgroup of the sandpile group is a given -group , is proportional to . For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.
Our results extends a recent theorem of Huang saying that the adjacency matrices of random -regular directed graphs are invertible with high probability to the undirected case.
September 20, Hao Shen, UW-Madison
Title: Stochastic quantization of Yang-Mills
Abstract: "Stochastic quantization” refers to a formulation of quantum field theory as stochastic PDEs. Interesting progress has been made these years in understanding these SPDEs, examples including Phi4 and sine-Gordon. Yang-Mills is a type of quantum field theory which has gauge symmetry, and its stochastic quantization is a Yang-Mills flow perturbed by white noise. In this talk we start by an Abelian example where we take a symmetry-preserving lattice regularization and study the continuum limit. We will then discuss non-Abelian Yang-Mills theories and introduce a symmetry-breaking smooth regularization and restore the symmetry using a notion of gauge-equivariance. With these results we can construct dynamical Wilson loop and string observables. Based on [S., arXiv:1801.04596] and [Chandra,Hairer,S., work in progress].
September 27, Timo Seppäläinen UW-Madison
Title:Random walk in random environment and the Kardar-Parisi-Zhang class
Abstract:This talk concerns a relationship between two much-studied classes of models of motion in a random medium, namely random walk in random environment (RWRE) and the Kardar-Parisi-Zhang (KPZ) universality class. Barraquand and Corwin (Columbia) discovered that in 1+1 dimensional RWRE in a dynamical beta environment the correction to the quenched large deviation principle obeys KPZ behavior. In this talk we condition the beta walk to escape at an atypical velocity and show that the resulting Doob-transformed RWRE obeys the KPZ wandering exponent 2/3. Based on joint work with Márton Balázs (Bristol) and Firas Rassoul-Agha (Utah).
October 4, Elliot Paquette, OSU
Title: Distributional approximation of the characteristic polynomial of a Gaussian beta-ensemble
Abstract: The characteristic polynomial of the Gaussian beta--ensemble can be represented, via its tridiagonal model, as an entry in a product of independent random two--by--two matrices. For a point z in the complex plane, at which the transfer matrix is to be evaluated, this product of transfer matrices splits into three independent factors, each of which can be understood as a different dynamical system in the complex plane. Conjecturally, we show that the characteristic polynomial is always represented as product of at most three terms, an exponential of a Gaussian field, the stochastic Airy function, and a diffusion similar to the stochastic sine equation. We explain the origins of this decomposition, and we show partial progress in establishing part of it.
Joint work with Diane Holcomb and Gaultier Lambert.
October 11, Chris Janjigian, University of Utah
Title: Busemann functions and Gibbs measures in directed polymer models on Z^2
Abstract: We consider the model of a nearest-neighbor random walk on the planar square lattice in a general iid space-time potential, which is also known as a directed polymer in a random environment. We prove results on existence, uniqueness (and non-uniqueness), and the law of large numbers for semi-infinite path measures. Our main tools are the Busemann functions, which are families of stochastic processes obtained through limits of ratios of partition functions.
Based on joint work with Firas Rassoul-Agha
October 18-20, Midwest Probability Colloquium, No Seminar
October 25, Promit Ghosal, Columbia
Title: Tails of the KPZ equation
Abstract: The KPZ equation is a fundamental stochastic PDE related to modeling random growth processes, Burgers turbulence, interacting particle system, random polymers etc. It is related to another important SPDE, namely, the stochastic heat equation (SHE). In this talk, we focus on the tail probabilities of the solution of the KPZ equation. For instance, we investigate the probability of the solution being smaller or larger than the expected value. Our analysis is based on an exact identity between the KPZ equation and the Airy point process (which arises at the edge of the spectrum of the random Hermitian matrices) and the Brownian Gibbs property of the KPZ line ensemble.
This talk will be based on a joint work with my advisor Prof. Ivan Corwin.
November 1, James Melbourne, University of Minnesota
Title: Upper bounds on the density of independent vectors under certain linear mappings
Abstract: Using functional analytic techniques and rearrangement, we prove anti-concentration results for the linear images of independent random variables, in the form of density upper bounds. For continuous variables the results unify and sharpen Bobkov-Chistyakov's for independent sums of vectors and Rudelson-Vershynin's bounds on projections of independent coordinates. For integer valued variables the techniques reduce finding the maximum of the probability mass function of a sum of independent variables, to the case that each variable is uniform on a contiguous interval. This problem is approached through analysis of characteristic functions and new $L^p$ bounds on the Dirichlet and Fejer Kernel are obtained and used to derive a discrete analog of Bobkov-Chistyakov.
November 8, Thomas Leblé, NYU
Title: The Sine-beta process: DLR equations and applications
Abstract: One-dimensional log-gases, or Beta-ensembles, are statistical physics models finding an incarnation in random matrix theory. Their limit behavior at microscopic scale is known as the Sine-beta process, its original description involves systems of coupled SDE's. In a joint work with D. Dereudre, A. Hardy, and M. Maïda, we give a new description of Sine-beta as an "infinite volume Gibbs measure", using the Dobrushin-Lanford-Ruelle (DLR) formalism, and we use it to prove the rigidity of the process, in the sense of Ghosh-Peres. Another application is a CLT for fluctuations of linear statistics.
November 22, Thanksgiving Break, No Seminar
Monday, November 26, 4pm, Van Vleck 911 Vadim Gorin, MIT
Title: Macroscopic fluctuations through Schur generating functions
Abstract: I will talk about a special class of large-dimensional stochastic systems with strong correlations. The main examples will be random tilings, non-colliding random walks, eigenvalues of random matrices, and measures governing decompositions of group representations into irreducible components.
It is believed that macroscopic fluctuations in such systems are universally described by log-correlated Gaussian fields. I will present an approach to handle this question based on the notion of the Schur generating function of a probability distribution, and explain how it leads to a rigorous confirmation of this belief in a variety of situations.
Wednesday, December 5 at 4pm in Van Vleck 911 Subhabrata Sen, MIT and Microsoft Research New England
Title: Random graphs, Optimization, and Spin glasses
Abstract: Combinatorial optimization problems are ubiquitous in diverse mathematical applications. The desire to understand their “typical” behavior motivates a study of these problems on random instances. In spite of a long and rich history, many natural questions in this domain are still intractable to rigorous mathematical analysis. Graph cut problems such as Max-Cut and Min-bisection are canonical examples in this class. On the other hand, physicists study these questions using the non-rigorous “replica” and “cavity” methods, and predict complex, intriguing features. In this talk, I will describe some recent progress in our understanding of their typical properties on random graphs, obtained via connections to the theory of mean-field spin glasses. The new techniques are broadly applicable, and lead to novel algorithmic and statistical consequences.