Eigenvalue degeneracy in sparse random matrices

TL;DR

The paper analyzes eigenvalue degeneracy in random matrices, showing zero degeneracy probability for matrices with absolutely continuous entries but positive degeneracy probability for sparse matrices with discontinuities at the origin. It blends discriminant-based results for continuous ensembles with Erdős–Rényi graph theory of perfect matchings to derive asymptotic probabilities of simple spectra in sparse regimes, obtaining that the probability of all eigenvalues being distinct tends to e^{-oldsymbol{ heta}} + oldsymbol{ heta} e^{-oldsymbol{ heta}} with oldsymbol{ heta} = 2 e^{-c} when the edge probability scales as p(n) = (\log n + c)/n. The symmetric case yields analogous results, reinforcing the link between spectral degeneracy and graph connectivity; degeneracy thus arises from eigenvalue accumulation at zero due to distribution discontinuities. These findings illuminate how spectral properties of sparse random matrices are governed by underlying graph structure, with potential implications for stability and dynamics in systems modeled by such 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 Erdös-Rényi 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.