# Probability Seminar

# Spring 2021

**Thursdays in 901 Van Vleck Hall at 2:30 PM**, unless otherwise noted.
**We usually end for questions at 3:20 PM.**

** IMPORTANT: ** In Spring 2021 the seminar is being run online. ZOOM LINK

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## January 28, 2021, no seminar

## February 4, 2021, Hong-Bin Chen (Courant Institute, NYU)

**Dynamic polymers: invariant measures and ordering by noise**

We develop a dynamical approach to infinite volume polymer measures (IVPM) in random environments. We define polymer dynamics in 1+1 dimension as a stochastic gradient flow, and establish ordering by noise. We prove that, for a fixed asymptotic slope, the polymer dynamics has a unique invariant distribution given by a unique IVPM. Moreover, One Force-One Solution principle holds.

## February 11, 2021, Kevin Yang (Stanford)

**Non-stationary fluctuations for some non-integrable models**

We will discuss recent progress on weak KPZ universality and non-integrable particle systems, including long-range models and slow bond models. The approach is based on a preliminary step in a non-stationary (first-order) Boltzmann-Gibbs principle. We will also discuss the full non-stationary Boltzmann-Gibbs principle itself and pieces of its proof.

## February 18, 2021, Ilya Chevyrev (Edinburgh)

**Signature moments to characterize laws of stochastic processes**

The normalized sequence of moments characterizes the law of any finite-dimensional random variable. In this talk, I will describe an extension of this result to path-valued random variables, i.e. stochastic processes, by using the normalized sequence of signature moments. I will show how these moments define a metric for laws of stochastic processes. This metric can be efficiently estimated from finite samples, even if the stochastic processes themselves evolve in high-dimensional state spaces. As an application, I will describe a non-parametric two-sample hypothesis test for laws of stochastic processes.

## February 25, 2021, Roger Van Peski (MIT)

**Random matrices, random groups, singular values, and symmetric functions**

Since the 1989 work of Friedman-Washington, the cokernels of random p-adic matrices drawn from various distributions have provided models for random finite abelian p-groups arising in number theory and combinatorics, the most famous being the class groups of quadratic imaginary number fields. Since any finite abelian p-group is isomorphic to a direct sum of cyclic groups $\bigoplus_i \mathbb{Z}/p^{\lambda_i}\mathbb{Z}$, it is equivalent to study the random integer partition $\lambda = (\lambda_1, \lambda_2,\ldots)$, which is analogous to the singular values of a complex random matrix. We show that the behavior of such partitions under taking products and corners of random p-adic matrices is governed by the Hall-Littlewood polynomials, recovering and explaining some previous results relating p-adic matrix cokernels to these polynomials. We use these exact results to study the joint asymptotic behavior of the cokernels of products of many random p-adic matrices $A_\tau \cdots A_1$, with $\tau$ acting as a discrete time parameter. We show that the parts $\lambda_i$ of the corresponding partition have a simple description via an interacting particle system, and their fluctuations converge under rescaling to independent Brownian motions. At both the exact and asymptotic level we explain connections between our results and existing results on singular values of complex random matrices: both are in fact degenerations of the same operations on random partitions coming from Macdonald polynomials.

## March 4, 2021, Roland Bauerschmidt (Cambridge)

**The Coleman correspondence at the free fermion point**

Two-dimensional statistical and quantum field theories are special in many ways. One striking instance of this is the equivalence of certain bosonic and fermionic fields, known as bosonization. I will first review this correspondence in the explicit instance of the massless Gaussian free field and massless Euclidean Dirac fermions. I will then present a result that extends this correspondence to the non-Gaussian `massless' sine-Gordon field on $R^2$ at $\beta=4\pi$ and massive Dirac fermions. This is an instance of Coleman's prediction that the `massless' sine-Gordon model and the massive Thirring model are equivalent. We use this correspondence to show that correlations of the `massless' sine-Gordon model decay exponentially for $\beta=4\pi$. This is joint work with C. Webb (arXiv:2010.07096).

## March 11, 2021, Sevak Mkrtchyan (Rochester)

**The limit shape of the Leaky Abelian Sandpile Model**

The leaky abelian sandpile model (Leaky-ASM) is a growth model in which n grains of sand start at the origin in the square lattice and diffuse according to a toppling rule. A site can topple if its amount of sand is above a threshold. In each topple a site sends some sand to each neighbor and leaks a portion 1-1/d of its sand. This is a dissipative generalization of the Abelian Sandpile Model, which corresponds to the case d=1.

We will discuss how, by connecting the model to a certain killed random walk on the square lattice, for any fixed d>1, an explicit limit shape can be computed for the region visited by the sandpile when it stabilizes.

We will also discuss the limit shape in the regime when the dissipation parameter d converges to 1 as n grows, as this is related to the ordinary ASM with a modified initial configuration.

## March 18, 2021, Theo Assiotis (Edinburgh)

**On the joint moments of characteristic polynomials of random unitary matrices**

I will talk about the joint moments of characteristic polynomials of random unitary matrices and their derivatives. In joint work with Jon Keating and Jon Warren we establish the asymptotics of these quantities for general real values of the exponents as the size N of the matrix goes to infinity. This proves a conjecture of Hughes from 2001. In subsequent joint work with Benjamin Bedert, Mustafa Alper Gunes and Arun Soor we focus on the leading order coefficient in the asymptotics, we connect this to Painleve equations for general values of the exponents and obtain explicit expressions corresponding to the so-called classical solutions of these equations.

## March 25, 2021, Wlodzimierz Bryc (Cincinnati)

**Fluctuations of particle density for open ASEP**

I will review results on fluctuations of particle density for the open Asymmetric Simple Exclusion Process. I will explain the statements and the Laplace transform duality arguments that appear in the proofs.

The talk is based on past and ongoing projects with Alexey Kuznetzov, Yizao Wang and Jacek Wesolowski.

## April 1, 2021, Zoe Huang (Duke University)

**Motion by mean curvature in interacting particle systems**

There are a number of situations in which rescaled interacting particle systems have been shown to converge to a reaction diffusion equation (RDE) with a bistable reaction term. These RDEs have traveling wave solutions. When the speed of the wave is nonzero, block constructions have been used to prove the existence or nonexistence of nontrivial stationary distributions. Here, we follow the approach in a paper by Etheridge, Freeman, and Pennington to show that in a wide variety of examples when the RDE limit has a bistable reaction term and traveling waves have speed 0, one can run time faster and further rescale space to obtain convergence to motion by mean curvature. This opens up the possibility of proving that the sexual reproduction model with fast stirring has a discontinuous phase transition, and that in Region 2 of the phase diagram for the nonlinear voter model studied by Molofsky et. al there were two nontrivial stationary distributions.

## April 8, 2021, Tianyi Zheng (UCSD)

**Random walks on wreath products and related groups**

Random walks on lamplighter groups were first considered by Kaimanovich and Vershik to provide examples of amenable groups with nontrivial Poisson boundary. Such processes can be understood rather explicitly, and provide guidance in the study of random walks on more complicated groups. In this talk we will discuss behavior of random walks on lamplighter groups, their extensions and some related groups which carry a similar semi-direct product structure.

## April 15, 2021, Keith Levin (UW-Madison, Statistics)

**Network Embeddings and Latent Space Models**

Networks are data structures that describe relations among entities, such as friendships among people in a social network or synapses between neurons in a brain. The field of statistical network analysis aims to develop network analogues of classical statistical techniques, and latent space models have emerged as the workhorse of this nascent field. Under these models, network formation is driven by unobserved geometric structure, in which each vertex in the network has an associated point in some metric space, called its latent position, that describes the (stochastic) behavior of the vertex in the network. In this talk, I will discuss some of my own work related to latent space models, focusing on 1) estimation of the vertex-level latent positions and 2) generating bootstrap replicates of network data. Throughout the talk, I will make a point to highlight open problems and ongoing projects that are likely to be of interest to probabilists.

## April 16, 2021, Matthew Junge (CUNY) FRIDAY at 2:25pm, joint with ACMS

**Modeling COVID-19 Spread in Universities**

University policy surrounding COVID-19 often involves big decisions informed by minimal data. Models are a tool to bridge this divide. I will describe some of the work that came out during Summer of 2020 to inform college reopening for Fall 2020. This includes a stochastic, agent-based model on a network for infection spread in residential colleges that I developed alongside a biologist, computer scientist, and group of students [1]. Time-permitting, I will describe a new project that aims to predict the impact of vaccination on infection spread in urban universities during the Fall 2021 semester. Disclaimer: I self-identify as a "pure" probabilist who typically proves theorems about particle systems [2]. These projects arose from my feeling compelled to help out to the best of my abilities during the height of the pandemic.

## April 22, 2021, Benjamin Fehrman (Oxford)

**Non-equilibrium fluctuations in interacting particle systems and conservative stochastic PDE**

Abstract: Interacting particle systems have found diverse applications in mathematics and several related fields, including statistical physics, population dynamics, and machine learning. We will focus, in particular, on the zero range process and the symmetric simple exclusion process. The large-scale behavior of these systems is essentially deterministic, and is described in terms of a hydrodynamic limit. However, the particle process does exhibit large fluctuations away from its mean. Such deviations, though rare, can have significant consequences---such as a concentration of energy or the appearance of a vacuum---which make them important to understand and simulate.

In this talk, which is based on joint work with Benjamin Gess, I will introduce a continuum model for simulating rare events in the zero range and symmetric simple exclusion process. The model is based on an approximating sequence of stochastic partial differential equations with nonlinear, conservative noise. The solutions capture to first-order the central limit fluctuations of the particle system, and they correctly simulate rare events in terms of a large deviations principle.

## April 29, 2021, James Martin (Oxford)

**The environment seen from a geodesic in last-passage percolation, and the TASEP seen from a second-class particle**

We study directed last-passage percolation in $\mathbb{Z}^2$ with i.i.d. exponential weights. What does a geodesic path look like locally, and how do the weights on and nearby the geodesic behave? We show convergence of the distribution of the "environment" as seen from a typical point along the geodesic in a given direction, as its length goes to infinity. We describe the limiting distribution, and can calculate various quantities such as the density function of a typical weight, or the proportion of "corners" along the path. The analysis involves a link with the TASEP (totally asymmetric simple exclusion process) seen from an isolated second-class particle, and we obtain some new convergence and ergodicity results for that process. The talk is based on joint work with Allan Sly and Lingfu Zhang.