Convergence of the logarithm of the characteristic polynomial of unitary Brownian motion to the Gaussian free field

Johannes Forkel, Isao Sauzedde

Published 2022-11-03Version 1

We prove that the real and imaginary parts of the logarithm of the characteristic polynomial of unitary Brownian motion converge to Gaussian free fields on the cylinder, as the matrix dimension goes to infinity. This is the natural dynamical analogue of the result for a fixed time by Hughes, Keating and O'Connell [1]. Further it complements a result by Spohn [2] on linear statistics of unitary Brownian motion, and a recent result by Bourgade and Falconet [3] connecting the characteristic polynomial of unitary Brownian motion to a Gaussian multiplicative chaos measure. 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.