# 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)

## 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.