Eigenvalue degeneracy in sparse random matrices

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.