EVENT DETAILS AND ABSTRACT

Title: The law of large numbers for the maximum of the log-potential of random matrices
Speaker: Elliot Paquette
Speaker Info: The Ohio State University
Brief Description:
Special Note:
Abstract:

The characteristic polynomials of random matrices are random functions that are anticipated to have many similar statistics to canonical log-correlated Gaussian random fields. We discuss some specific conjectures on the maxima of characteristic polynomials, and we show that the maximum of the centered log-modulus of the characteristic polynomial of GUE (the Gaussian Unitary Ensembles) is log N + o(log N) with high probability. This confirms the first term in a conjecture by Fyodorov and Simm. Moreover, we prove a general theorem about 'almost' Gaussian fields that should be applicable to showing the law of large numbers for the log characteristic polynomial of beta-ensembles. We will also survey some extensions of this type of result: to other potentials, to other beta, and to higher degrees of precision. This is based on joint works with Gaultier Lambert and Ofer Zeitouni.