Zongrui Yang

Zongrui Yang

Math PhD student

Talks and presentations

TA session for the minicourse Invariant Measures for Exclusion Processes by Dominik Schmid

July 15, 2024

TA session, [Virginia Integrable Probability Summer School 2024](https://sites.google.com/view/dominik-schmid/virginia-2024), University of Virginia

In this mini-course, we investigate invariant measures for exclusion processes. After a brief introduction to some classical results on the set of invariant measures for the simple exclusion processes on infinite graphs, we turn our focus to the asymmetric simple exclusion process with open boundaries, also called the open ASEP. We introduction the matrix product ansatz, and show how it can be used to study the stationary distribution of the open ASEP. We discuss different representations of the matrix product ansatz. Using a particular solution for the matrix product ansatz with Askey-Wilson polynomials, we then introduce (signed) Askey-Wilson measures, and explain how the stationary distribution of the open ASEP can be used to construct stationary solutions to the open Kardar-Parisi-Zhang equation. In the last lecture, we focus on some very recent developments for the special case of the open TASEP: a so-called two-layer Gibbs representation of its invariant measure.

Askey–Wilson signed measures and their applications in open ASEP

May 04, 2024

Talk, [6th Cincinnati Symposium on Probability](https://homepages.uc.edu/~wangyz/CSP2024/index.html), University of Cincinnati

We introduce the Askey–Wilson signed measures as a new tool for studying the stationary measure of the open asymmetric simple exclusion process (ASEP). As applications of this technique, we access several asymptotics of open ASEP: the density profile, limit fluctuations, large deviations, open KPZ fixed-point limit, half-line ASEP limit, and open ASEP with a light particle. Based on joint works with Zhipeng Liu, Dominik Schmid, Yizao Wang and Jacek Wesolowski.

Askey-Wilson signed measures and open ASEP

April 26, 2024

Talk, [University of Chicago Probability and Statistical Physics Seminar (Spring 2024)](https://math.uchicago.edu/~probstatphys/), University of Chicago

We introduce the Askey-Wilson signed measures as a new tool for studying the stationary measure of the open asymmetric simple exclusion process (ASEP). As applications of this technique, we derive several asymptotic properties of open ASEP: the density profile, limit fluctuations, the open KPZ fixed-point limit, the half-line ASEP limit, large deviations, and open ASEP with light particles. Based on joint works with Zhipeng Liu, Dominik Schmid, Yizao Wang and Jacek Wesolowski.

Stationary measures for integrable models with two open boundaries

February 29, 2024

Talk, [UW Madison Probability Seminar (Spring 2024)](https://wiki.math.wisc.edu/index.php/Probability_Seminar), University of Wisconsin-Madison

We present two methods to study the stationary measures of integrable systems with two open boundaries. The first method is based on Askey-Wilson signed measures, which is illustrated for the open asymmetric simple exclusion process and the six-vertex model on a strip. The second method is based on two-layer Gibbs measures and is illustrated for the geometric last-passage percolation and log-gamma polymer on a strip. This talk is based on joint works with Yizao Wang, Jacek Wesolowski, Guillaume Barraquand and Ivan Corwin.

Stationary measures for integrable models with two open boundaries

January 16, 2024

Talk, [Probability Seminar at Warsaw University of Technology](https://pages.mini.pw.edu.pl/~semiproba/streszczenia.htm#s2024-01-16), Online

We present two methods to study the stationary measures of integrable systems with two open boundaries. The first method is based on Askey-Wilson signed measures, which is illustrated for the open asymmetric simple exclusion process and the six-vertex model on a strip. The second method is based on two-layer Gibbs measures and is illustrated for the geometric last-passage percolation and log-gamma polymer on a strip. This talk is based on two works: arXiv:2307.06574 with Yizao Wang and Jacek Wesolowski and arXiv:2306.05983 with Guillaume Barraquand and Ivan Corwin.

Askey–Wilson signed measures and open ASEP in the shock region

January 05, 2024

Talk, [AMS-AWM Special Session on Solvable Lattice Models and their Applications Associated with the Noether Lecture](https://meetings.ams.org/math/jmm2024/meetingapp.cgi/Symposium/428), The Moscone Center, San Francisco, CA

The open asymmetric simple exclusion process (ASEP) is a paradigmatic model for non-equilibrium systems with open boundaries and for KPZ universality. The phase diagram consists of the fan region and the shock region. In this talk, we present a new method for studying the stationary measure of open ASEP in the shock region. We introduce a family of multi-dimensional Askey–Wilson signed measures and describe the joint generating function of the stationary measure in terms of integrations with respect to these Askey–Wilson signed measures. As an application, we offer a rigorous derivation of the density profile and limit fluctuations of open ASEP in the shock region, confirming the existing physics postulations.

Stationary measures for integrable polymers on a strip

December 05, 2023

Talk, [Penn/Temple Probability Seminar (Fall 2023)](https://www.math.upenn.edu/events/stationary-measures-integrable-polymers-strip), University of Pennsylvania

We prove that the stationary measures for the geometric last passage percolation (LPP) and log-gamma polymer models on a diagonal strip are given by the marginals of objects we call two-layer Gibbs measures. Taking an intermediate disorder limit of the log-gamma polymer stationary measure, we recover the conjectural description of the open KPZ equation stationary measure for all choices of boundary parameters. This is a joint work with Guillaume Barraquand and Ivan Corwin.

Askey-Wilson signed measures and open ASEP in the shock region

October 06, 2023

Talk, [The Asymmetric Simple Exclusion Process: October 2-6, 2023](https://scgp.stonybrook.edu/archives/39876), Simons Center for Geometry and Physics

In this talk, we will present a new method for studying the stationary measure of the open asymmetric simple exclusion process (ASEP) in the shock region. We introduce a family of multi-dimensional Askey-Wilson signed measures and then describe the joint generating function of the open ASEP stationary measure in terms of integrations with respect to these Askey-Wilson signed measures. As an application, we offer a rigorous derivation of the density profile and limit fluctuations of the open ASEP in the shock region, confirming the existing physics postulations. This is a joint work with Yizao Wang and Jacek Wesolowski.

Stationary measure for six-vertex model on a strip

March 29, 2023

Talk, [Taft Research Seminar on Probability Theory and its Applications (Spring 2023)](https://homepages.uc.edu/~brycwz/Taft-2023.html), University of Cincinnati

In this talk we will study a family of Markov chains governed by the six-vertex model on a strip. Their stationary measure can be solved by the matrix product ansatz and then characterized by the Askey-Wilson process. We study the limit of mean particle density and obtain the phase diagram.