On the Spectrum of Random Anti-symmetric and Tournament Matrices

TL;DR

This paper studies the eigenvalue distribution of a class of non-Hermitian random matrices related to anti-symmetric matrices, revealing interlacing properties and extending Hermitian tools to anti-symmetric cases.

Contribution

It introduces a novel random matrix model combining anti-symmetric matrices with rank-one perturbations and analyzes their eigenvalue behavior.

Findings

Eigenvalues' real parts are asymptotically interlaced with those of the anti-symmetric matrix.

Tools from Hermitian matrix analysis extend to anti-symmetric matrices.

Eigenvalue distributions exhibit predictable asymptotic behavior.

Abstract

We consider a discrete, non-Hermitian random matrix model, which can be expressed as a shift of a rank-one perturbation of an anti-symmetric matrix. We show that, asymptotically almost surely, the real parts of the eigenvalues of the non-Hermitian matrix around any fixed index remain interlaced with those of the anti-symmetric matrix. Along the way, we show that some tools recently developed to study the eigenvalue distributions of Hermitian matrices extend to the anti-symmetric setting.