The law of large numbers for the maximum of almost Gaussian log-correlated fields coming from random matrices

TL;DR

This paper determines the asymptotic behavior of the maximum of a log-correlated field derived from random matrices, confirming a conjecture and introducing a method based on second-moment bounds and hyperbolic branching.

Contribution

It verifies the leading order of a conjecture for the maximum of log-correlated fields from random matrices using a novel second-moment method and hyperbolic branching structure analysis.

Findings

Confirmed the conjecture for GUE matrices.

Developed a method comparing fields to Gaussian models via exponential moments.

Applicable to other point process-derived fields with computable exponential moments.

Abstract

We compute the leading asymptotics as $N\to\infty$ of the maximum of the field $Q_N(q)= \log\det|q- A_N|$, $q\in \mathbb{C}$, for any unitarily invariant Hermitian random matrix $A_N$ associated to a non-critical real-analytic potential. Hence, we verify the leading order in a conjecture of Fyodorov and Simm formulated for the GUE. The method relies on a classical upper-bound and a more sophisticated lower-bound based on a variant of the second-moment method which exploits the hyperbolic branching structure of the field $Q_N(q)$, $q$ in the upper half plane. Specifically, we compare $Q_N$ to an idealized Gaussian field by means of exponential moments. In principle, this method could also be applied to random fields coming from other point processes provided that one can compute certain mixed exponential moments. For unitarily invariant ensembles, we show that these assumptions follow from the Fyodorov-Strahov formula and asymptotics of orthogonal polynomials derived by Deift, Kriecherbauer, McLaughlin, Venakides, and Zhou.