Seminar time: Thursdays 10:20-11:15am.

**January 14, 2021: **Roger Van Peski (MIT)

**Title: **Limits and fluctuations of p-adic random matrix products

**Abstract: **Any nonsingular matrix A over the p-adic integers Z_p may, by Smith normal form, be written as U diag(p^{\lambda_1},\ldots,p^{\lambda_n}) V for U, V in GL_n(Z_p), for some integers \lambda_i called *singular numbers* (also known as invariant factors or elementary divisors). The distributions of singular numbers for random A are quite well-studied, largely because the equivalent data of the abelian p-group coker(A) provides useful models for many random groups in number theory and combinatorics, collectively called Cohen-Lenstra heuristics.

By contrast, in this talk we consider a setting in which the (rescaled) fluctuations of singular numbers converge to continuous probability distributions on R. Namely, we show that the singular numbers of successive products A_k \cdots A_1, where A_i are corners of independent Haar-distributed p-adic matrices, obey a law of large numbers and their fluctuations converge dynamically to independent Brownian motions with k playing the role of time. These asymptotics rely on new exact expressions for distributions of singular numbers of products and corners of random matrices over Q_p in terms of Hall-Littlewood polynomials, which allow us to re-express the above matrix product process as a simple interacting particle system. We also indicate parallels between our results–both exact and asymptotic–and known results on singular values of complex matrices.

**January 21, 2021: **Vadim Gorin (Wisconsin)

**Title: **Infinite beta random matrix theory

**Abstract: **Dyson’s threefold approach suggests to deal with real/complex/quaternion random matrices as beta=1/2/4 instances of beta-ensembles. We complement this approach by the beta=\infty point, whose study reveals a number of previously unnoticed algebraic structures. Our central object is the G\inftyE ensemble, which is a counterpart of the classical Gaussian Orthogonal/Unitary/Symplectic ensembles. We encounter unusual orthogonal polynomials, random walks, and finite free polynomial convolutions.

**January 28, 2021: **Lisa Sauermann (IAS/MIT)

**Title: **On polynomials that vanish to high order on most of the hypercube

**Abstract: **Motivated by higher vanishing multiplicity generalizations of Alon’s Combinatorial Nullstellensatz and its applications, we study the following problem: for fixed k and n large with respect to k, what is the minimum possible degree of a polynomial P in R[x_1,…,x_n] such that P(0,…,0) is non-zero and such that P has zeroes of multiplicity at least k at all points in {0,1}^n except the origin? For k=1, a classical theorem of Alon and Füredi states that the minimum possible degree of such a polynomial equals n. We solve the problem for all k>1, proving that the answer is n+2k−3.

Joint work with Yuval Wigderson.

**February 4, 2021: **Charles Bordenave (Institut de Mathématiques de Marseille)

**Title: **Entropy of processes on infinite trees

**Abstract: ** We define a natural notion of micro-state entropy associated to a random process on a unimodular random tree. This entropy is closely related to Bowen’s sofic entropy in dynamical systems. It is also connected to factor models on random graphs. We give a formula for this entropy for a large class of processes.

This is a joint work with Agnes Backhausz et Balasz Szegedy.

**February 11, 2021: **David Sivakoff (OSU)

**Title: **Neighborhood growth in heterogeneous environments

**Abstract: **The simplest deterministic growth dynamics on the vertices of Z^d are the bootstrap percolation, or threshold growth models, wherein an initially occupied set is enlarged by iteratively including vertices that have at least r>0 occupied neighbors. Initially occupied vertices are chosen independently with probability p, and we are interested in the time at which the origin is first occupied (which may be infinite). I will survey some of the vast literature on this problem and its generalizations, and discuss recent progress in heterogeneous environments.

**February 18, 2021: **Leonid Petrov (UVA)

**Title: **Solvable directed polymer models

**Abstract: **I will discuss integrable random polymers (based on gamma / inverse gamma or beta distributed weights), and explain their connection to known and new families of symmetric functions. There is some interesting combinatorics which is yet to be discovered.

**February 25, 2021: **Richard Kenyon (Yale)

**Title: **The multitiling model

**Abstract: **The study of random tilings is a cornerstone area of combinatorics and probability. Unfortunately the tiling problem is NP-hard even in quite simple-looking cases. We study a tractable variant, the multitiling model, where we tile a region with high multiplicity. In the limit of large multiplicities we compute the asymptotic growth rate of the number of multitilings: the free energy of the multitiling model. We show that the individual tile densities tend to a Gaussian field with respect to an associated discrete Laplacian. For tilings with translates of a polyomino on Z^2 we find crystallization phenomena (and accompanying phase transitions), and even naturally occurring quasicrystals.

This is joint work with Andrei Pohoata (Yale).

**March 4, 2021: **Anton Bernshteyn (Georgia Tech)

**Title: **Lower bounds for difference bases

**Abstract: **A difference basis with respect to $n$ is a subset $A \subseteq \mathbb{Z}$ such that $A – A \supseteq [n]$. R\'{e}dei and R\'{e}nyi showed that the minimum size of a difference basis with respect to $n$ is $(c+o(1))\sqrt{n}$ for some positive constant $c$. The precise value of $c$ is not known, but some bounds are available, and I will discuss them in this talk. This is joint work with Michael Tait (Villanova University).

**March 11, 2021: **Hugo Duminil-Copin (IHES)

**Title: **Emerging symmetries in 2D percolation

**Abstract: **A great achievement of physics in the second half of the twentieth century has been the prediction of conformal symmetry of the scaling limit of critical statistical physics systems. Around the turn of the millennium, the mathematical understanding of this fact progressed tremendously in two dimensions with the introduction of the Schramm-Loewner Evolution and the proofs of conformal invariance of the Ising model and dimers. Nevertheless, the understanding is still restricted to very specific models. In this talk, we will gently introduce the notion of conformal invariance of lattice systems by taking the example of percolation models. We will also explain some recent proof of rotational invariance for a large class of such models. This represents an important progress in the direction of proving full conformal invariance.

**March 18, 2021: **Matthew Kwan (Stanford)

**Title: **Permanents of Random Symmetric matrices

**Abstract: **In this talk we discuss some recent work joint with Lisa Sauermann, proving that the permanent of a uniformly random symmetric n×n matrix with ±1 entries typically has magnitude n^{n/2 + o(n)}. This resolves a conjecture of Vu.

**March 25, 2021: **Gabriel Conant (University of Cambridge)

**Title: **Quantitative stable arithmetic regularity in arbitrary finite groups

**Abstract: **In 2011, Malliaris and Shelah showed that “stable” finite graphs admit a strong form of Szemeredi’s regularity lemma with polynomial bounds and no irregular pairs. Here “stable” is a combinatorial property, motivated by model theory, which is defined by omitting so-called “half-graphs”. In 2017, Terry and Wolf developed an analogue of this result for subsets of finite abelian groups, which is based on the notion of arithmetic regularity as invented by Green. Roughly speaking, they showed that any stable subset of a finite abelian group can be efficiently approximated by cosets of a subgroup whose index is bounded exponentially in the approximation error and the stability constant. Around the same time, in joint work with Pillay and Terry, we proved a version of this for arbitrary finite groups using model theoretic techniques, which resulted in stronger qualitative features, but lacked any information about quantitative bounds. In this talk, I will discuss a new effective proof of our result, which yields quantitative bounds for arbitrary finite groups, and improves the bound in Terry & Wolf’s result from exponential to polynomial.

**March 25, 2021 (3-3:55pm, joint with Analysis seminar): **Yen Do (UVA)

**Title: **Real roots of random orthogonal polynomials

**Abstract: **We consider linear combinations of orthonormal polynomials (up to some degree n) with respect to a fixed compactly supported measure on the real line, and the linear coefficients are independent random variables with zero mean and unit variance. Such a combination is also known as a random orthogonal polynomial. Under mild technical assumptions, we establish universality for the leading asymptotics (as n go to infinity) of the average number of real roots of the random orthogonal polynomial, both globally and locally. This is joint work with Oanh Nguyen (UIUC) and Van Vu (Yale).

**April 1, 2021: **Sivaguru Sritharan (Air Force Institute of Technology, Dayton)

**Title: **Stochastic Navier-Stokes Equations: Ergodicity, Large Deviations, Control and Filtering

**Abstract: **In this talk we will give an overview of several rigorous studies and results over the past number of years on compressible and incompressible Navier-Stokes and Euler equations including martingale, pathwise and strong solutions, invariant measures and ergodicity, large deviations of small noise (Freidlin-Wentzell) and large time (Donsker-Varadhan) type, filtering and control and related infinite dimensional partial differential equations, optimal stopping and impulse control and related infinite dimensional variational and quasi-variational inequalities. We will also outline developments in related subjects such as magnetohydrodynamics as well as PDEs in physics such as the Einstein field equation, Maxwell-Dirac equation and the nonlinear Schrodinger equation, all subject to Gaussian and Levy noise.

**April 8, 2021: **Marcus Michelen (UIC)

**Title: **Roots of random polynomials near the unit circle

**Abstract: **It is a well-known (but perhaps surprising) fact that a polynomial with independent random coefficients has most of its roots very close to the unit circle. Using a probabilistic perspective, we understand the behavior of roots of random polynomials exceptionally close to the unit circle and prove several limit theorems; these results resolve several conjectures of Shepp and Vanderbei. We will also discuss how our techniques provide a heuristic, probabilistic explanation for why random polynomials tend to have most roots near the unit circle. Based on joint work with Julian Sahasrabudhe.

**April 15, 2021: **Michael Tate (Villanova University)

**Title: **Spectral Tur\’an problems

**Abstract: **In this talk we will discuss what subgraphs can be guaranteed if a graph has a large eigenvalue. This is the spectral analog of the Tur\’an problem and was first raised by Brualdi and Solheid and Nikiforov. We will give an overview of how to prove theorems in this area and will discuss some intuition for how to guess what the extremal graph(s) should be. This is joint work with Sebi Cioaba, Dheer Desai, Lihua Feng, Josh Tobin, and Xiao-Dong Zhang.

**April 15, 2021 (11:30-12:25): **Patrick Devlin (Yale University)

**Title:** “Buying Votes to (Probably) Win a Random Election”

**Abstract: **In the low-stakes world of Duckburg politics, Scrooge McDuck is running for office, and he only has one opponent. Each morning leading up to the election, all the citizens of Duckburg wake up, they check social media, and they notice who their friends are planning to vote for. Then in the evening, everybody updates their profiles so that their new political opinion agrees with the majority of what their friends thought in the morning. [This updating process is called “majority dynamics”]

Senator McDuck isn’t above greasing a few palms to buy some votes, but he is a notoriously tight-fisted miser who refuses to bribe more voters than he needs to. He doesn’t know who’s actually friends with whom, so ultimately he’ll have to leave it up to chance. In this talk, we explore how many voters he needs to pay off in order for him to be 99% sure that he’ll end up unanimously winning the election, and we’ll also discuss how long it will take the citizens of Duckburg to reach some sort of consensus. Our main result is a central limit theorem for how many voters will favor McDuck after one day assuming the underlying graph is drawn from the Erd\H{o}s-R\’enyi model. This is joint work with Ross Berkowitz.

**April 22, 2021: **Leonardo T. Rolla (Warwick)

**Title:**Activated random walks on Zd

**Abstract: **Some stochastic systems are particularly interesting as they exhibit critical behavior without fine-tuning of a parameter, a phenomenon called self-organized criticality. In the context of driven-dissipative steady states, one of the main models is that of Activated Random Walks. Long-range effects intrinsic to its conservative dynamics and lack of a simple algebraic structure cause standard tools and techniques to break down, which makes the mathematical study of this model remarkably challenging. Yet, some exciting progress has been made in the last ten years, with the development of a framework of tools and methods which is finally becoming more structured. We will briefly recall the existing results and open problems, then focus on recent progress for one-dimensional symmetric walks with at density (Basu-Ganguly-Hoffman), enhancement and continuity of the critical curve (Taggi), scaling limit at criticality (Cabezas-myself), symmetric walks at high sleep rate (Hoffman-Richey-myself), and linear growth (Levine-Silvestri).

**April 29, 2021: **Vincent Tassion (ETH)

**Title: **Crossing probabilities for planar percolation

**Abstract: **Percolation models were originally introduced to describe the propagation of a fluid in a random medium. In dimension two, the percolation properties of a model are encoded by so-called crossing probabilities (probabilities that certain rectangles are crossed from left to right). In the eighties, Russo, Seymour and Welsh obtained general bounds on crossing probabilities for Bernoulli percolation (the most studied percolation model, where edges of a lattice are independently erased with some given probability 1-p). These inequalities rapidly became central tools to analyze the critical behavior of the model. In this talk I will present a new result which extends the Russo–Seymour–Welsh theory to general percolation measures satisfying two properties: symmetry and positive correlation.

This is a joint work with Laurin Köhler-Schindler