Probability Seminar

Back to Probability Group

Past Seminars

Spring 2023

Thursdays at 2:30 PM either in 901 Van Vleck Hall or on Zoom

We usually end for questions at 3:20 PM.

ZOOM LINK. Valid only for online seminars.

If you would like to sign up for the email list to receive seminar announcements then please join our group.

January 26, 2023, in person: Evan Sorensen (UW-Madison)

The stationary horizon as a universal object for KPZ models

The last 5-10 years has seen remarkable progress in constructing the central objects of the KPZ universality class, namely the KPZ fixed point and directed landscape. In this talk, I will discuss a third central object known as the stationary horizon (SH). The SH is a coupling of Brownian motions with drifts, indexed by the real line, and it describes the unique coupled invariant measures for the directed landscape. I will talk about how the SH appears as the scaling limit of several models, including Busemann processes in last-passage percolation and the TASEP speed process. I will also discuss how the SH helps to describe the collection of infinite geodesics in all directions for the directed landscape. Based on joint work with Timo Seppäläinen and Ofer Busani.

February 2, 2023, in person: Jinsu Kim (POSTECH)

Fast and slow mixing of continuous-time Markov chains with polynomial rates

Continuous-time Markov chains on infinite positive integer grids with polynomial rates are often used in modeling queuing systems, molecular counts of small-size biological systems, etc. In this talk, we will discuss continuous-time Markov chains that admit either fast or slow mixing behaviors. For a positive recurrent continuous-time Markov chain, the convergence rate to its stationary distribution is typically investigated with the Lyapunov function method and canonical path method. Recently, we discovered examples that do not lend themselves easily to analysis via those two methods but are shown to have either fast mixing or slow mixing with our new technique. The main ideas of the new methodologies are presented in this talk along with their applications to stochastic biochemical reaction network theory.

February 9, 2023, in person: Jeffrey Kuan (Texas A&M)

Shift invariance for the multi-species q-TAZRP on the infinite line

We prove a shift--invariance for the multi-species q-TAZRP (totally asymmetric zero range process) on the infinite line. Similar-looking results had appeared in works by [Borodin-Gorin-Wheeler] and [Galashin], using integrability, but are on the quadrant. The proof in this talk relies instead on a combinatorial approach, in which the state space is generalized to a poset, and the totally asymmetric process is generalized to a monotone process on a poset. The continuous-time process is decomposed into its discrete embedded Markov chain and its exponential holding times, and the shift-invariance is proved using explicit contour integral formulas. Open problems about multi-species ASEP will be discussed as well.

February 16, 2023, in person: Milind Hegde (Columbia)

Understanding the upper tail behaviour of the KPZ equation via the tangent method

The Kardar-Parisi-Zhang (KPZ) equation is a canonical non-linear stochastic PDE believed to describe the evolution of a large number of planar stochastic growth models which make up the KPZ universality class. A particularly important observable is the one-point distribution of its analogue of the fundamental solution, which has featured in much of its recent study. However, in spite of significant recent progress relying on explicit formulas, a sharp understanding of its upper tail behaviour has remained out of reach. In this talk we will discuss a geometric approach, related to the tangent method introduced by Colomo-Sportiello and rigorously implemented by Aggarwal for the six-vertex model. The approach utilizes a Gibbs resampling property of the KPZ equation and yields a sharp understanding for a large class of initial data.

February 23, 2023, in person: Swee Hong Chan (Rutgers)

Log-concavity and cross product inequalities in order theory

Given a finite poset that is not completely ordered, is it always possible find two elements x and y, such that the probability that x is less than y in the random linear extension of the poset, is bounded away from 0 and 1? Kahn-Saks gave an affirmative answer and showed that this probability falls between 3/11 (0.273) and 8/11 (0.727). The currently best known bound is 0.276 and 0.724 by Brightwell-Felsner-Trotter, and it is believed that the optimal bound should be 1/3 and 2/3, also known as the 1/3-2/3 Conjecture. Most notably, log-concave and cross product inequalities played the central role in deriving both bounds. In this talk we will discuss various generalizations of these results together with related open problems. This talk is joint work with Igor Pak and Greta Panova, and is intended for the general audience.