Probability Seminar

Elliot PaquetteMcGill University
The random matrix Fyodorov-Hiary-Keating conjecture.

Monday, March 13, 2023 - 3:45pm
Malott 406

Abstract: The Fyodorov-Hiary-Keating conjecture has two parts, one in random matrix theory and one about the Riemann zeta function. In the random matrix part, it gives the precise distributional limit for the maximum of a characteristic polynomial of a Haar Unitary matrix. Using the replica method and a physically motivated `freezing’ ansatz, they derived one of the most precise log-correlated field predictions to date, and they did it for a process which was not even Gaussian. While existing work shows that Haar Unitary matrices had many log correlated field connections, techniques for showing convergence of the maximum typically rely on either the Gaussianity of the underlying process or precise branching structures built into the problem; the characteristic polynomial has neither. We will describe the problem and the current state of the art, in which we (the speaker and Ofer Zeitouni) show the convergence in law of the maximum of a Circular-beta ensemble random matrix to a convolution of a gumbel and the total mass of a (non-Gaussian) critical multiplicative chaos.