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

Walid Hachem, Michail Louvaris

TL;DR

This work analyzes the spectral radius of large non-Hermitian random matrices with a general variance profile. By studying the reverse characteristic polynomial $q_n(z)$ and establishing its asymptotic equivalence to a random analytic function $\kappa_n(z)\exp(-F_n(z))$, the authors derive a high-probability confinement $\rho(X^{(n)}) \le \sqrt{\rho(S^{(n)})}$ (under suitable normalizations) and extend the analysis to sparse and continuous variance profiles. The approach, inspired by Bordenave–Chafaï–García-Zelada, requires only minimal moment assumptions and yields examples including block profiles, continuous kernels, and inhomogeneous directed Erdős–Rényi models, with robust results under both dense and sparse regimes. These findings link the spectral properties of random matrices to the geometry of their variance profiles, enabling applications to dynamical systems, networks, and related stochastic models where inhomogeneity and sparsity are essential features.

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.

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

TL;DR

This work analyzes the spectral radius of large non-Hermitian random matrices with a general variance profile. By studying the reverse characteristic polynomial and establishing its asymptotic equivalence to a random analytic function , the authors derive a high-probability confinement (under suitable normalizations) and extend the analysis to sparse and continuous variance profiles. The approach, inspired by Bordenave–Chafaï–García-Zelada, requires only minimal moment assumptions and yields examples including block profiles, continuous kernels, and inhomogeneous directed Erdős–Rényi models, with robust results under both dense and sparse regimes. These findings link the spectral properties of random matrices to the geometry of their variance profiles, enabling applications to dynamical systems, networks, and related stochastic models where inhomogeneity and sparsity are essential features.

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.
Paper Structure (18 sections, 18 theorems, 123 equations)

This paper contains 18 sections, 18 theorems, 123 equations.

Table of Contents

  1. Problem description and results
  2. Case studies
  3. Block variance profile
  4. Sampling a continuous variance profile
  5. Random sparse sampling à la Erdős–Rényi of a continuous variance profile
  6. Random sparse sampling of a continuous variance profile with a fixed outer degree
  7. Proof of Theorems \ref{['th-main']} and \ref{['th-chpol']}
  8. Proof of Theorem \ref{['th-chpol']}: preliminary results on random holomorphic functions.
  9. Tightness of $(q_n)$
  10. Asymptotics of the finite-dimensional distributions when $W_{11}$ is bounded
  11. Proof of Theorem \ref{['th-chpol']} when $W_{11}$ is bounded
  12. Releasing the boundedness assumption on $W_{11}$. End of the proof of Theorem \ref{['th-chpol']}
  13. Proof of Theorem \ref{['th-main']} from Theorem \ref{['th-chpol']}
  14. Proofs for Section \ref{['sec-app']}
  15. Proof of Proposition \ref{['TC']}
  16. ...and 3 more sections

Key Result

Theorem 1.3

Let Assumptions bnd-S and bnd-det hold true. Then, it holds that

Theorems & Definitions (30)

  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • Proposition 2.3
  • Corollary 2.4
  • Remark 2.5
  • Remark 2.6
  • ...and 20 more