On the spectral radius and the characteristic polynomial of a random matrix with independent elements and a variance profile

TL;DR

This paper establishes that the spectral radius of large non-Hermitian random matrices with general variance profiles is asymptotically bounded by the spectral radius of the variance profile matrix, extending understanding of spectral properties under minimal assumptions.

Contribution

It proves a new bound on the spectral radius for matrices with general variance profiles and demonstrates asymptotic equivalence of the characteristic polynomial to a random analytic function.

Findings

Spectral radius does not exceed the square root of the variance profile's spectral radius with high probability.

Asymptotic equivalence between the reverse characteristic polynomial and a random analytic function.

Applicable to matrices with piecewise constant, continuous variance profiles, and inhomogeneous Erdős-Rényi models.

Abstract

In this paper, it is shown that with large probability, the spectral radius of a large non-Hermitian random matrix with a general variance profile does not exceed the square root of the spectral radius of the variance profile matrix. A minimal moment assumption is considered and sparse variance profiles are covered. Following an approach developed recently by Bordenave, Chafa{\"i} and Garc{\'i}a-Zelada, the key theorem states the asymptotic equivalence between the reverse characteristic polynomial of the random matrix at hand and a random analytic function which depends on the variance profile matrix. The result is applied to the case of a non-Hermitian random matrix with a variance profile given by a piecewise constant or a continuous non-negative function, the inhomogeneous (centered) directed Erd\H{o}s-R{\'e}nyi model, and more.