Spring 2023

January 19:

January 26:

February 2: Ander Aguirre (Ohio State University)

Title:  CLTs for Pair Dependent Statistics of Circular Beta Ensembles

Abstract:  In this talk we give an overview of recent results on the fluctuation of the statistic \sum_{i\neq j} f(L_N(\theta_i-\theta_j)) for the Circular Beta Ensemble in the global, mesoscopic and local regimes. This work is morally related to Johansson’s 1988 CLT for the linear statistic \sum_i f(\theta_i) and Lambert’s subsequent 2019 extension to the mesoscopic regime. The special case of the CUE (\beta=2) in the local regime L_N=N is motivated by Montgomery’s study of pair correlations of the rescaled zeros of the Riemann zeta function. Our techniques are of combinatorial nature for the CUE and analytical for \beta\neq2.

 

February 9:

February 16: Hugo Falconet (NYU)

Title: Liouville quantum gravity from random matrix dynamics

Abstract:  The Liouville quantum gravity measure is a properly renormalized exponential of the 2d GFF. In this talk, I will explain how it appears as a limit of natural random matrix dynamics: if (U_t) is a Brownian motion on the unitary group at equilibrium, then the measures $|det(U_t – e^{i theta}|^gamma dt dtheta$ converge to the 2d LQG measure with parameter $gamma$, in the limit of large dimension. This extends results from Webb, Nikula and Saksman for fixed time. The proof relies on a new method for Fisher-Hartwig asymptotics of Toeplitz determinants with real symbols, which extends to multi-time settings. I will explain this method and how to obtain multi-time loop equations by stochastic analysis on Lie groups. 

 
Based on a joint work with Paul Bourgade.

February 23:

March 2 (Arnold Classic – Hotels are scarce):

March 9: Sumit Mukherjee (Columbia University)

Title: Asymptotic Distribution of Quadratic Forms

Abstract: In this talk we will give an exact characterization for the asymptotic distribution of quadratic forms in IID random variables with finite second moment, where the underlying matrix is the adjacency matrix of a graph. In particular we will show that the limit distribution of such a quadratic form can always be expressed as the sum of three independent components: a Gaussian, a (possibly) infinite sum of centered chi-squares, and a Gaussian with a random variance. As a consequence, we derive necessary and sufficient conditions for asymptotic normality, and universality of the limiting distribution. 

 

This talk is based on joint work with B. B. Bhattacharya, S. Das, and S. Mukherjee.

 

 

March 16: Spring Break (no talks)

March 23: Chris Hoffman (U. Washington)

March 30: Roger Van Peski (MIT)

April 6:

April 13: Brian Rider (Temple University)

April 20: Xiaoyu Dong (University of Michigan)

April 27: Colin Defant (MIT) [Seminar will be held at 2pm in CH 232]

Title: Ungarian Markov Chains
Abstract: Inspired by Ungar’s solution to the famous slopes problem, we introduce Ungar moves, which are operations that can be performed on elements of a finite lattice L. Applying Ungar moves randomly results in an absorbing Markov chain that we call the Ungarian Markov chain of L. For a variety of interesting lattices L, we focus on estimating E(L), the expected number of steps of this Markov chain needed to get from the top element of L to the bottom element of L. When L is distributive, its Ungarian Markov chain is equivalent to an instance of the well-studied random process known as last-passage percolation with geometric weights. One of our main results states that if L is a trim lattice, then E(L) is at most E(spine(L)), where spine(L) is a specific distributive sublattice of L called the spine of L. Combining this lattice-theoretic theorem with known results about last-passage percolation yields a powerful method for proving upper bounds for E(L) when L is trim. This talk is based on joint work with Rupert Li.