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Exponential growth of random determinants beyond invariance

Gérard Ben Arous, Paul Bourgade and Benjamin McKenna

Vol. 3 (2022), No. 4, 731–789
Abstract

We give simple criteria to identify the exponential order of magnitude of the absolute value of the determinant for wide classes of random matrix models, not requiring the assumption of invariance. These include Gaussian matrices with covariance profiles, Wigner matrices and covariance matrices with subexponential tails, Erdős–Rényi and d-regular graphs for any polynomial sparsity parameter, and non-mean-field random matrix models, such as random band matrices for any polynomial bandwidth. The proof builds on recent tools, including the theory of the matrix Dyson equation as developed by Ajanki, Erdős, and Krüger (2019).

We use these asymptotics as an important input to identify the complexity of classes of Gaussian random landscapes in the companion papers by Ben Arous, Bourgade, and McKenna (2023+) and McKenna (2023+).

Keywords
determinant, random matrix, Kac–Rice, matrix Dyson equation, Wigner matrix, Erdős–Rényi, d-regular, band matrix, Wishart matrix, free addition
Mathematical Subject Classification
Primary: 60B20
Milestones
Received: 12 May 2021
Revised: 15 April 2022
Accepted: 13 June 2022
Published: 21 February 2023
Authors
Gérard Ben Arous
Courant Institute of Mathematical Sciences
New York University
New York, NY
United States
Paul Bourgade
Courant Institute of Mathematical Sciences
New York University
New York, NY
United States
Benjamin McKenna
Courant Institute of Mathematical Sciences
New York University
New York, NY
United States