Exponential growth of random determinants beyond invariance

TL;DR

This paper establishes criteria for determining the exponential growth rate of the absolute value of determinants across various non-invariant random matrix models, broadening understanding beyond classical invariance assumptions.

Contribution

It introduces simple criteria for exponential order of determinants applicable to diverse random matrices, including non-invariant models like band matrices and sparse graphs.

Findings

Determinants exhibit exponential growth in broad classes of random matrices.

The criteria apply to Gaussian, Wigner, Erdős-Rényi, and band matrices.

Asymptotics inform complexity analysis of Gaussian landscapes in related work.

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, Erd\H{o}s-R\'enyi 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 in [Ajanki, Erd\H{o}s, Kr\"uger 2019]. We use these asymptotics as an important input to identify the complexity of classes of Gaussian random landscapes in our companion papers [Ben Arous, Bourgade, McKenna 2021; McKenna 2021].