# Probability and Mathematical Physics Seminar

#### Distributional approximation of the characteristic polynomial of a Gaussian beta-ensemble

**Speaker:**
Elliot Paquette, Ohio State

**Location:**
Warren Weaver Hall 512

**Date:**
Friday, October 26, 2018, 11 a.m.

**Synopsis:**

The characteristic polynomial of the Gaussian beta-ensemble can be represented, via its tridiagonal model, as an entry in a product of independent random two-by-two matrices. For a point z in the complex plane, at which the transfer matrix is to be evaluated, this product of transfer matrices splits into three independent factors, each of which can be understood as a different dynamical system in the complex plane. Conjecturally, we show that the characteristic polynomial is always represented as product of at most three terms, an exponential of a Gaussian field, the stochastic Airy function, and a diffusion similar to the stochastic sine equation.

We explain the origins of this decomposition, and we show partial progress in establishing part of it.

Joint work with Diane Holcomb and Gaultier Lambert.