Large-$N$ Eigenvalue Distribution of Randomly Perturbed Asymmetric Matrices

TL;DR

This paper derives the eigenvalue density distribution for large asymmetric matrices perturbed by random noise, revealing bounds on the support area and expressing the density in terms of the unperturbed matrix's eigenvalues.

Contribution

It provides a large-$N$ limit analysis of eigenvalue distributions for perturbed asymmetric matrices, including bounds on support and explicit formulas when commuting conditions are met.

Findings

Eigenvalue density is bounded in the large-$N$ limit.

Support area of the eigenvalue density cannot be less than πv².

Explicit density expression when the unperturbed matrix commutes with its conjugate.

Abstract

The density of complex eigenvalues of random asymmetric $N\times N$ matrices is found in the large-$N$ limit. The matrices are of the form $H_0+A$ where $A$ is a matrix of $N^2$ independent, identically distributed random variables with zero mean and variance $N^{-1}v^2$. The limiting density $\rho (z,z^*)$ is bounded. The area of the support of $\rho (z,z^*)$ cannot be less than $\pi v^2$. In the case of $H_0$ commuting with its conjugate, $\rho (z,z^*)$ is expressed in terms of the eigenvalue distribution of the non-perturbed part $H_0$.