# Probability Seminar

# Fall 2021

**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.

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## September 16, 2021, in person: Hanbaek Lyu (UW-Madison)

**Scaling limit of soliton statistics of a multicolor box-ball system**

The box-ball systems (BBS) are integrable cellular automata whose long-time behavior is characterized by the soliton solutions, and have rich connections to other integrable systems such as Korteweg-de Veris equation. Probabilistic analysis of BBS is an emerging topic in the field of integrable probability, which often reveals novel connection between the rich integrable structure of BBS and probabilistic phenomena such as phase transition and invariant measures. In this talk, we give an overview on the recent development in scaling limit theory of multicolor BBS with random initial configurations. Our analysis uses various methods such as modified Greene-Kleitman invariants for BBS, circular exclusion processes, Kerov–Kirillov–Reshetikhin bijection, combinatorial R, and Thermodynamic Bethe Ansatz.

## September 23, 2021, no seminar

## September 30, 2021, in person: Marianna Russkikh (MIT)

**Lozenge tilings and the Gaussian free field on a cylinder**

We discuss new results on lozenge tilings on an infinite cylinder, which may be analyzed using the periodic Schur process introduced by Borodin. Under one variant of the $q^{vol}$ measure, corresponding to random cylindric partitions, the height function converges to a deterministic limit shape and fluctuations around it are given by the Gaussian free field in the conformal structure predicted by the Kenyon-Okounkov conjecture. Under another variant, corresponding to an unrestricted tiling model on the cylinder, the fluctuations are given by the same Gaussian free field with an additional discrete Gaussian shift component. Fluctuations of the latter type have been previously conjectured for tiling models on planar domains with holes.

## October 7, 2021, ZOOM: Barbara Dembin (ETH Zurich)

**The time constant for Bernoulli percolation is Lipschitz continuous strictly above $p_c$**

We consider the standard model of i.i.d. first passage percolation on $\mathbb Z^d$ given a distribution $G$ on $[0,+\infty]$ ($+\infty$ is allowed). When $G([0,+\infty))>p_c(d)$, it is known that the time constant $\mu_G$ exists. We are interested in the regularity properties of the map $G\mapsto\mu_G$. We study the specific case of distributions of the form $G_p=p\delta_1+(1-p)\delta_\infty$ for $p>p_c(d)$. In this case, the travel time between two points is equal to the length of the shortest path between the two points in a bond percolation of parameter $p$. We prove that the function $p\mapsto \mu_{G_p}$ is Lipschitz continuous on every interval $[p_0,1]$, where $p_0>p_c(d)$. This is a joint work with Raphaël Cerf.

## October 14, 2021, UPDATED FORMAT: ZOOM: Evan Sorensen (UW-Madison)

**Busemann functions and semi-infinite geodesics in a semi-discrete space**

In the last 10-15 years, Busemann functions have been a key tool for studying semi-infinite geodesics in planar first and last-passage percolation. We study Busemann functions in the semi-discrete Brownian last-passage percolation (BLPP) model and use these to derive geometric properties of the full collection of semi-infinite geodesics in BLPP. This includes a characterization of uniqueness and coalescence of semi-infinite geodesics across all asymptotic directions. To deal with the uncountable set of points in BLPP, we develop new methods of proof and uncover new phenomena, compared to discrete models. For example, for each asymptotic direction, there exists a random countable set of initial points out of which there exist two semi-infinite geodesics in that direction. Further, there exists a random set of points, of Hausdorff dimension ½, out of which, for some random direction, there are two semi-infinite geodesics that split from the initial point and never come back together. We derive these results by studying variational problems for Brownian motion with drift.

## October 21, 2021, ZOOM: Sumit Mukherjee (Columbia)

**Fluctuations in Mean Field Ising Models**

We study fluctuations of the magnetization (average of spins) in an Ising model on a sequence of "well-connected" approximately $d_n$ regular graphs on $n$ vertices. We show that if $d_n\gg n^{1/2}$, then the fluctuations are universal, and same as that of the Curie–Weiss model, in the entire ferromagnetic parameter regime. We then give a counterexample to show that $d_n\gg n^{1/2}$ is actually tight, in the sense that the limiting distribution changes if $d_n\sim n^{1/2}$ except in the high temperature regime. By refining our argument, we show that in the high temperature regime universality holds for $d_n\gg n^{1/3}$. As a by-product of our proof technique, we prove rates of convergence, as well as exponential concentration for the sum of spins, and tight estimates for several statistics of interest.

This is based on joint work with Nabarun Deb at Columbia University.

## October 28, 2021, ZOOM: Wei-Kuo Chen (Minnesota)

**Grothendieck $L_p$ problem for Gaussian matrices**

The Grothendieck $L_p$ problem is defined as an optimization problem that maximizes the quadratic form of a Gaussian matrix over the unit $L_p$ ball. The $p=2$ case corresponds to the top eigenvalue of the Gaussian Orthogonal Ensemble, while for $p=\infty$ this problem is known as the ground state energy of the Sherrington-Kirkpatrick mean-field spin glass model and its limit can be expressed by the famous Parisi formula. In this talk, I will describe the limit of this optimization problem for general $p$ and discuss some results on the behavior of the near optimizers along with some open problems. This is based on a joint work with Arnab Sen.

## November 4, 2021, ZOOM: Mackenzie Simper (Stanford)

**Double Cosets, Mallows Measure, and a Transvections Markov Chain**

If $G = GL_n(\mathbb{F}_q)$ and $B$ is the subgroup of lower triangular matrices, then the $B\backslash G/B$ double cosets are indexed by permutations $S_n$. This is the famous Bruhat decomposition, closely related to the LU decomposition of a matrix. The Markov chain on $G$ generated by random transvections – matrices which fix a hyperplane – induces a Markov chain on $S_n$ with the Mallows measure as stationary distribution. We characterize this process, study the mixing time, and discuss the connection with the number of pivoting steps needed in Gaussian elimination. This is joint work with Persi Diaconis and Arun Ram.

## November 11, 2021, in person: Lingfu Zhang (Princeton)

**Shift-invariance of the colored TASEP and random sorting networks**

In this talk, I will introduce a new shift-invariance property of the colored TASEP. It is in a similar spirit as recent results for the six-vertex models (by Borodin-Gorin-Wheeler and Galashin), but its proof is via non-algebraic arguments. This new shift-invariance is applied to prove a conjectured distributional equality between the classical exponential Last Passage Percolation model and the oriented swap process (OSP). The OSP is a model for a random sorting network, with N particles labeled $1,\dots,N$ performing successive adjacent swaps at random times until they reach the reverse configuration $N,\dots,1$. Our distributional equality implies new asymptotic results about the OSP, some of which are in connection with the Airy sheet.

## November 18, 2021, in person: Mariya Shcherbina: (Kharkov)

**Supersymmetric approach to the deformed Ginibre ensemble.**

We consider non Hermitian random matrices of the form $H=A+H_0$, where $A$ is a rather general $n\times n$ matrix (Hermitian or non-Hermitian) independent of $H_0$, and $H_0$ is a standard Ginibre matrix. It is known that under some reasonable conditions the limiting spectrum of $H$ makes some domain $D$ with a smooth boundary $\Gamma$. We apply the supersymmetric approach to study the behavior of the smallest singular value of the matrix $(H-z)$ if $z\in D$ but $dist(z,\Gamma)\sim N^{-1/2}$.

## November 25, 2021, no seminar

## December 2, 2021, in person: Xuan Wu (Chicago)

## December 9, 2021, ZOOM: Sunil Chhita (Durham)

**GOE Fluctuations for the maximum of the top path in ASMs**

The six-vertex model is an important toy-model in statistical mechanics for two-dimensional ice with a natural parameter Δ. When Δ=0, the so-called free-fermion point, the model is in natural correspondence with domino tilings of the Aztec diamond. Although this model is integrable for all Δ, there has been very little progress in understanding its statistics in the scaling limit for other values. In this talk, we focus on the six-vertex model with domain wall boundary conditions at Δ=1/2, where it corresponds to alternating sign matrices (ASMs). We consider the level lines in a height function representation of ASMs. We report that the maximum of the topmost level line for a uniformly random ASMs has the GOE Tracy-Widom distribution after appropriate rescaling. This talk is based on joint work with Arvind Ayyer and Kurt Johansson.