Eigenvalue degeneracy in sparse random matrices

TL;DR

This paper investigates how eigenvalue degeneracy occurs in sparse random matrices with discontinuous entries, revealing a positive degeneracy probability due to eigenvalue accumulation at the origin, contrasting with continuous cases.

Contribution

It introduces a probabilistic analysis of eigenvalue degeneracy in sparse matrices with discontinuous entries using graph theory methods.

Findings

Positive degeneracy probability due to eigenvalue accumulation at zero

Degeneracy probability differs from continuous matrix cases

Asymptotic evaluation using Erdös-Rényi bipartite graph theory

Abstract

In random matrices with independent and continuous matrix entries, the degeneracy probability of the eigenvalues is known to be zero. In this paper, random matrices including discontinuous matrix entries are analyzed in order to observe how degeneracy is generated. Using Erd\"os-R\'enyi matching probability theory of random bipartite graphs, we asymptotically evaluate the degeneracy probability of such random matrices. As a result, due to accumulation of the eigenvalues to the origin, a positive degeneracy probability is found for eigenvalues of a sparse random matrix model.