# Probability Seminar

# 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.

## March 2, 2023, in person: Max Hill (UW-Madison)

**On the Effect of Intralocus Recombination on Triplet-Based Species Tree Estimation**

My talk will introduce some key topics in mathematical phylogenetics and is intended to be accessible for those not familiar with the field. I will discuss joint work with Sebastien Roch on the subject of species tree estimation from multiple loci subject to intralocus recombination. The focus is on R*, a summary coalescent-based method using rooted triplets. I will present a result showing how intralocus recombination can give rise to an "inconsistency zone," in which correct inference using R* is not assured even in the limit of infinite amount of data.

## March 9, 2023, in person: Xuan Wu (U. Chicago)

**From the KPZ equation to the directed landscape**

This talk presents the convergence of the KPZ equation to the directed landscape, which is the central object in the KPZ universality class. This convergence result is the first to the directed landscape among the positive temperature models.

## March 23, 2023, in person: Jiaming Xu (UW-Madison)

**Rectangular Matrix addition in low and high temperatures**

We study the addition of two [math]\displaystyle{ {\scriptsize M \times N} }[/math] rectangular random matrices with certain invariant distributions in two limit regimes, where the parameter [math]\displaystyle{ {\scriptsize \beta} }[/math] (inverse temperature) goes to infinity and zero. In low temperature regime the random singular values of the sum concentrate at deterministic points, while in high temperature regime we obtain a Law of Large Numbers of the empirical measures. Our proof uses the so-called type BC Bessel function as characteristic function of rectangular matrices, and through the analysis of this function we introduce a new family of cumulants, that linearize the addition in high temperature limit, and degenerate to the classical or free cumulants in special cases.

## March 30, 2023, in person: Bálint Virág (Toronto)

**The planar stochastic heat equation and the directed landscape**

The planar stochastic heat equation describes heat flow or random polymers on an inhomogeneous surface. It is a finite-temperature version of planar first passage percolation such as the Eden growth model. It is the first model with plane symmetries for which we can show convergence to the directed landscape. The methods use a Skorokhod integral representation and Gaussian multiplicative chaos on path space.

Joint work with Jeremy Quastel and Alejandro Ramirez.

## April 6, 2023, in person: Shankar Bhamidi (UNC-Chapel Hill)

**Disorder models for random graphs, Erdos’s leader problem, and power of limited choice models for network evolution**

First passage percolation, and more generally the study of diffusion of material through disordered systems is a fundamental area in probabilistic combinatorics with a vast body of work especially in the context of spatial systems.

The goal of this talk is to survey a slightly different setting for such questions namely the more “mean-field” setting of random graph models. We will describe the state of the art of this field, with the final goal of describing one of the main conjectures in this area namely the conjectured scaling limit of the minimal spanning tree and its dependence on the degree exponent of the corresponding network model. We will describe recent progress in this area, its connection to questions in dynamic network models, in particular Erdos’s leader problem for the identity of the maximal component for critical random graphs, and the intuition for understanding the evolution of maximal components through the critical scaling window from a different area of probabilistic combinatorics, namely the study of limited choice models for network evolution.