UCSD Probability Seminar, 2017-2018

The probability seminar meets at 10am on Thursdays in AP&M 6402, unless specifically indicated otherwise. Please send any inquiries to the organizer

T. Zheng ( tzheng2 at math dot ucsd dot edu.)

Spring 2018

á  Thursday April 19, 2018, 10am

Nick Cook, UCLA

Title: The maximum of the characteristic polynomial for a random permutation matrix

Abstract: Let $P$ be a uniform random permutation matrix of size $N$ and let $\chi_N(z)= \det(zI - P)$ denote its characteristic polynomial. We prove a law of large numbers for the maximum modulus of $\chi_N$ on the unit circle, specifically,

$\sup_{|z|=1}|\chi_N(z)|= N^{x_c + o(1)}$

with probability tending to one as $N\to \infty$, for a numerical constant $x_c\approx 0.677$. The main idea of the proof is to uncover an approximate branching structure in the distribution of (the logarithm of) $\chi_N$, viewed as a random field on the circle, and to adapt a well-known second moment argument for the maximum of the branching random walk. Unlike the well-studied \emph{CUE field} in which $P_N$ is replaced with a Haar unitary, the distribution of $\chi_N(z)$ is sensitive to Diophantine properties of the argument of $z$. To deal with this we borrow tools from the Hardy--Littlewood circle method in analytic number theory. Based on joint work with Ofer Zeitouni.

á  Thursday May 3, 2018, 10am

Lucian Beznea, IMAR

Title: Invariant, super and quasi-martingale functions of a Markov process

Abstract: We identify the linear space spanned by the real-valued excessive functions of a Markov process with the set of those functions which are quasimartingales when we compose them with the process. Applications to semi-Dirichlet forms are given. We provide a unifying result which clarifies the relations between harmonic, co-harmonic, invariant, co-invariant, martingale and co-martingale functions, showing that in the conservative case they are all the same. The talk is based on joint works with Iulian Cimpean (Bucharest) and Michael Roeckner (Bielefeld).

á  Thursday May 10, 2018, 10am

Pascal Maillard, UniversitŽ Paris-Sud

Title: Fluctuations of the Gibbs measure of branching Brownian motion at critical temperature

Abstract:  Branching Brownian motion is a prototype of a disordered system and a toy model for spin glasses and log-correlated fields. It also has an exact duality relation with the FKPP equation, a well-known reaction diffusion equation. In this talk, I will present recent results (obtained with Michel Pain) on the fluctuations of the Gibbs measure at the critical temperature. By Gibbs measure I mean here the measure whose atoms are the positions of the particles, weighted by their Gibbs weight. It is known that this Gibbs measure, after a suitable scaling, converges to a deterministic measure. We prove a non-standard central limit theorem for the integral of a function against the Gibbs measure, for a large class of functions. The possible limits are 1-stable laws with arbitrary asymmetry parameter depending on the function. In particular, the derivative martingale and the usual additive martingale satisfy such a central limit theorem with, respectively, a totally asymmetric and a Cauchy distribution.

á  Thursday May 31, 2018, 10am

Ian Charlesworth, UCSD

Title: Bi-free probability and an approach to conjugate variables.

Abstract: Free entropy theory is an analogue of information theory in a non-commutative setting, which has had great applications to the examination of structural properties of von Neumann algebras. I will discuss some ongoing joint work with Paul Skoufranis to extend this approach to the setting of bi-free probability which attempts to study simultaneously left'' and right'' non-commutative variables. I will speak in particular of an approach to a bi-free Fisher information and bi-free conjugate variables -- analogues of Fisher's information measure and the score function of information theory. The focus will be on constructing these tools in the non-commutative setting, and time permitting, I will also mention some results such as bi-free Cramer-Rao and Stam inequalities, and some quirks of the bi-free setting which are not present in the free setting.

á  Thursday June 7, 2018, 10am

Dan Romik, UC Davis

Title: Rational probabilities of connectivity events in loop percolation and fully packed loops

Abstract: In this talk I will describe a family of events arising in two related probability models, one having to do with uniformly random "fully packed loops" (a family of combinatorial objects which are in bijection with alternating sign matrices), and another appearing in connection with a natural random walk on noncrossing matchings. The connection between the two models is highly nonobvious and was conjectured by physicists Razumov and Stroganov in 2001, and given a beautiful proof in 2010 by Cantini and Sportiello. Another intriguing phenomenon is that the probabilities of the events in question, known as "connectivity events", appear to be rational functions of a size parameter N (with the simplest such formula being 3(N^2-1)/2(4N^2+1)), but this is only conjectured in all but a few cases. The attempts to prove such formulas by myself and others have led to interesting algebraic results on a family of multivariate polynomials known as "wheel polynomials", and to a family of conjectural constant term identities that is of independent interest and poses an interesting challenge to algebraic combinatorialists.

á  Thursday June 14, 2018, 10am

Brian Hall, University of Notre Dame

Title: Eigenvalues for Brownian motion in the general linear group

Abstract is here.

Winter 2018

á  Thursday February 8, 2018, 10am

Yumeng Zhang, Stanford University

Title: Rapid mixing of Glauber dynamics on hypergraph independent set

Abstract: Independent sets in hypergraphs can be encoded as 0-1 configurations on the set of vertices such that each hyperedge is adjacent to at least one 0. This model has been studied in the CS community for its large gap between efficient MCMC algorithms (previously d <(k-1)/2) and the conjectured onset of computational hardness (d > O(2^{k/2}) ), where d is the largest degree of vertices and k is the minimum size of hyperedges. In this talk we use a percolation approach to show that the Glauber dynamics is rapid mixing for d < O(2^{k/2} ), closing the gap up to a multiplicative constant.

This is joint work with Jonathan Hermon and Allan Sly.

á  Thursday February 15, 2018, 10am

Leonid Petrov, University of Virginia

Title: Nonequilibrium particle systems in inhomogeneous space

Abstract: I will discuss stochastic interacting particle systems in the KPZ universality class evolving in one-dimensional inhomogeneous space. The inhomogeneity means that the speed of a particle depends on its location. I will focus on integrable examples of such systems, i.e., for which certain observables can be written in exact form suitable for asymptotic analysis. Examples include a continuous-space version of TASEP (totally asymmetric simple exclusion process), and the pushTASEP (=long-range TASEP). For integrable systems, density limit shapes can be described in an explicit way. We also obtain asymptotics of fluctuations, in particular, around slow bonds and infinite traffic jams caused by slowdowns.

á  Thursday March 8, 2018, 10am

Georg Menz, UCLA

Title:  A quantitative theory of the hydrodynamic limit

Abstract: The hydrodynamic limit of the Kawasaki dynamics states that a certain stochastic evolution of a lattice system converges macroscopically to a deterministic non-linear heat equation. We will discuss how the statement of the hydrodynamic limit can be made quantitative. The key step is to introduce an additional evolution on a mesoscopic scale that emerges from projecting the microscopic observables onto splines. The hydrodynamic limit is then deduced in two steps. In the first step one shows the convergence of the microscopic to the mesoscopic evolution and in the second step one deduces the convergence of the mesoscopic to the macroscopic evolution.

The talk is about a joint work with Deniz Dizdar, Felix Otto and Tianqi Wu.

á  Thursday March 15, 2018, 10am

Karl Liechty, De Paul University

Title: Tacnode processes, winding numbers, and Painleve II

Abstract: Abstract: I will discuss a model of nonintersecting Brownian bridges on the unit circle, which produces quite a few universal determinantal processes as scaling limits. I will focus on the tacnode process, in which two groups of particles meet at a single point in space-time before separating, and introduce a new version of the tacnode process in which a finite number of particles "switch sides" before the two groups separate. We call this new process the k-tacnode process, and it is defined by a kernel expressed in terms of a system of tau-functions for the Painleve II equation. Technically, our model of nonintersecting Brownian bridges on the unit circle is studied using a system of discrete orthogonal polynomials with a complex (non-Hermitian) weight, so I'll also discuss some of the analytical obstacles to that analysis.
This is joint work with Dong Wang and Robert Buckingham

Fall 2017

á Thursday, Oct 5, 2017, 10am

Jean-Dominique Deuschel, TU Berlin

Title: Random walks in dynamical balanced environment

Abstract: We prove a quenched invariance principle and local limit theorem for a random walk in an ergodic balanced time dependent environment on the lattice. Our proof relies on the parabolic Harnack inequality for the adjoint operator. This is joint work with X. Guo.

á Thursday, Oct 12, 2017, 10am

Pierre-Olivier Goffard, UC Santa Barbara

Title: Boundary Crossing Problems with Applications to Risk Management.

Abstract: Many problems in stochastic modeling come down to study the crossing time of a certain stochastic process through a given boundary, lower or upper. Typical fields of application are in risk theory, epidemic modeling, queueing, reliability and sequential analysis. The purpose of this talk is to present a method to determine boundary crossing probabilities linked to stochastic point processes having the order statistic property. A very well-known boundary crossing result is revisited, a detailed proof is given. the same arguments may be used to derive results in trickier situations. We further discuss the practical implications of this classical.

á Thursday, Oct 19, 2017, 10am

Omer Tamuz, Caltech

Title: Large deviations in social learning

Abstract: Models of information exchange that originate from economics provide interesting questions in probability. We will introduce some of these models, discuss open questions, and explain some recent results.

Joint with Wade Hann-Caruthers, Matan Harel, Vadim Martynov, Elchanan Mossel and Philipp Strack

á Thursday, Nov 2, 2017, 10am

Qiang Zeng, Northwestern University

Title: The Sherrington-Kirkpatrick model is Full-step Replica Symmetry Breaking at zero temperature

Abstract: Starting in 1979, the physicist Giorgio Parisi wrote a series of ground breaking papers introducing the idea of replica symmetry breaking, which allowed him to predict a solution for the Sherrington-Kirkpatrick (SK) model by breaking the symmetry of replicas infinitely many times. This is known as full-step replica symmetry breaking (FRSB). In this talk, we will provide a mathematical proof of Parisi's FRSB prediction at zero temperature for the more general mixed p-spin model. More precisely, we will show that the functional order parameter of this model is not a step function. This talk is based on joint work with Antonio Auffinger and Wei-Kuo Chen.

á  Thursday, Nov 30, 2017, 10am

Tom Alberts, University of Utah

Title: Geometric Methods for Last Passage Percolation

Abstract:  In an attempt to generalize beyond solvable methods of analysis for last passage percolation, recently Eric Cator (Radboud University, Nijmegen) and I have started analyzing the piecewise linearity of the last passage model. The tools we use to this point are purely geometric, but have the potential advantage that they can be used for very general choices of random inputs. I will describe the very pretty geometry of the last passage model, our work in progress to use it to produce probabilistic information, and some connections to algebraic geometry.

á  Thursday, Dec 7, 2017, 10am

Anas Rahman, University of Melbourne

Title: Random Matrices and Loop Equations

Abstract: I will begin by introducing the Gaussian, Laguerre and Jacobi ensembles and their corresponding eigenvalue densities. The moments of these eigenvalue densities are generated by the corresponding resolvent, R(x). When investigating large matrices of size N, it is natural to expand R(x) as a series in 1/N, as N tends to infinity. The loop equation formalism enables one to compute R(x) to any desired order in 1/N via a triangular recursive system. This formalism is equivalent to the topological recursion, the Schwinger-Dyson equations and the Virasoro constraints, among other things. The loop equations provide a relatively accessible entry-point to these topics and my derivation will rely on nothing more than integration by parts, as Aomoto applied to the Selberg integral. Time permitting, I may also explore links to the topological recursion and/or some combinatorics.

All original results will be from joint work with Peter Forrester and Nicholas Witte.