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arXiv · 2501.03657

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

Walid Hachem·Michail Louvaris
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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{ï} and Garc{í}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 Erdős-R{é}nyi model, and more.

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Walid Hachem, Michail Louvaris. 2025-10-09. On the spectral radius and the characteristic polynomial of a random matrix with independent elements and a variance profile. https://arxiv.org/abs/2501.03657

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