Analytic approach for the number statistics of non-Hermitian random matrices

TL;DR

This paper develops an analytic method to determine the eigenvalue count statistics inside any contour for large non-Hermitian random matrices, applicable even without explicit eigenvalue distributions, demonstrated on asymmetric graph adjacency matrices.

Contribution

The paper introduces a general analytic framework for eigenvalue counting statistics in non-Hermitian matrices, applicable to complex ensembles lacking explicit joint eigenvalue distributions.

Findings

Derived expressions for mean and variance of eigenvalue counts.

Established a rate function for rare fluctuations of eigenvalue counts.

Validated theoretical predictions with numerical simulations showing excellent agreement.

Abstract

We introduce a powerful analytic method to study the statistics of the number $\mathcal{N}_{\textbf{A}}(\gamma)$ of eigenvalues inside any contour $\gamma \in \mathbb{C}$ for infinitely large non-Hermitian random matrices ${\textbf A}$. Our generic approach can be applied to different random matrix ensembles, even when the analytic expression for the joint distribution of eigenvalues is not known. We illustrate the method on the adjacency matrices of weighted random graphs with asymmetric couplings, for which standard random-matrix tools are inapplicable. The main outcome is an effective theory that determines the cumulant generating function of $\mathcal{N}_{\textbf{A}}$ via a path integral along $\gamma$, with the path probability distribution following from the solution of a self-consistent equation. We derive the expressions for the mean and the variance of $\mathcal{N}_{\textbf{A}}$ as well as for the rate function governing rare fluctuations of ${\mathcal{N}}_{\textbf{A}}{(\gamma)}$. All theoretical results are compared with direct diagonalization of finite random matrices, exhibiting an excellent agreement.