The eigenvalue spectrum of a large real antisymmetric random matrix with non-zero mean

TL;DR

This paper analyzes the eigenvalue spectrum of large real antisymmetric random matrices, revealing a semicircular distribution at zero mean and eigenvalue splitting at critical non-zero means, supported by numerical simulations.

Contribution

It introduces a fermionic and replica approach to derive the eigenvalue spectrum, including the effects of non-zero mean, which is a novel analytical contribution.

Findings

Semicircular eigenvalue spectrum at zero mean

Eigenvalues split off at critical non-zero mean values

Analytical results agree with numerical simulations

Abstract

We study the eigenvalue spectrum of a large real antisymmetric random matrix $J_{ij}$. Using a fermionic approach and replica trick, we obtain a semicircular spectrum of eigenvalues when the mean value of each matrix element is zero, and in the case of a non-zero mean, we show that there is a set of critical finite mean values above which eigenvalues arise that are split off from the semicircular continuum of eigenvalues. The result converged with numerical simulations.