Convergence of the logarithm of the characteristic polynomial of unitary Brownian motion in Sobolev space

TL;DR

This paper proves that the logarithm of the characteristic polynomial of unitary Brownian motion converges to Gaussian free fields in Sobolev spaces as matrix size increases, extending previous fixed-time results.

Contribution

It establishes convergence in Sobolev spaces for the dynamical case, and introduces a Wick-type identity of independent interest.

Findings

Convergence of logarithm of characteristic polynomial to Gaussian free fields.

Convergence holds in certain Sobolev spaces, believed to be optimal.

A new Wick-type identity is proved.

Abstract

We prove that the convergence of the real and imaginary parts of the logarithm of the characteristic polynomial of unitary Brownian motion toward Gaussian free fields on the cylinder, as the matrix dimension goes to infinity, holds in certain suitable Sobolev spaces, which we believe to be optimal. This is the natural dynamical analogue of the result for a fixed time by Hughes, Keating and O'Connell [1]. A weak kind of convergence is known since the work of Spohn [2], which was widely improved recently by Bourgade and Falconet [3]. In the course of this research we also proved a Wick-type identity, which we include in this paper, as it might be of independent interest.