Real spectra of large real asymmetric random matrices

TL;DR

This paper investigates the distribution of real eigenvalues in large real asymmetric random matrices, revealing a proportional relationship with the density of complex eigenvalues and applying it to various matrix ensembles.

Contribution

It establishes a universal relation between real and complex eigenvalue densities in large asymmetric matrices, enabling calculations across different ensembles.

Findings

Real eigenvalue density is proportional to the square root of complex eigenvalue density.

Derived a method to compute real eigenvalue densities using complex eigenvalue distributions.

Applied the theory to heavy-tailed and sparse random regular graph matrices.

Abstract

When a randomness is introduced at the level of real matrix elements, depending on its particular realization, a pair of eigenvalues can appear as real or form a complex conjugate pair. We show that in the limit of large matrix size the density of such real eigenvalues is proportional to the square root of the asymptotic density of complex eigenvalues continuated to the real line. This relation allows one to calculate the real densities up to a normalization constant, which is then applied to various examples, including heavy-tailed ensembles and adjacency matrices of sparse random regular graphs.