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

2016/11/27 by Gaultier Lambert, Lambert, Gaultier, Elliot Paquette +1
Mathematics · #Advanced Algebra and Geometry #FOS: Mathematics #Geometry and complex manifolds #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1611.08885

openalex publication_date 2016/11/27 · openalex created_date 2022/08/17 · openalex updated_date 2026/07/28

Abstract

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

Citations

• Random matrices: Universality of local eigenvalue statistics

Related

• Extrema of log-correlated random variables: Principles and Examples
• The maximum of log-correlated Gaussian fields in random environments
• Theory of random matrices with strong level confinement
• Fisher-Hartwig Asymptotics and Log-Correlated Fields in Random Matrix Theory
• Maximum of a log-correlated Gaussian field