Sparse Random Matrices have Simple Spectrum

TL;DR

This paper proves that certain classes of sparse random matrices and Erdős-Rényi graph adjacency matrices typically have simple, non-repeating eigenvalues, with results optimal in the sparsity exponent.

Contribution

It establishes the simple spectrum property for sparse symmetric random matrices and Erdős-Rényi graphs, extending understanding of eigenvalue multiplicities in sparse regimes.

Findings

Sparse random matrices have simple spectrum with high probability.

Erdős-Rényi graph adjacency matrices also have simple spectrum in specified sparsity range.

Results are optimal in the exponent of the sparsity parameter.

Abstract

Let $M_n$ be a class of symmetric sparse random matrices, with independent entries $M_{ij} = \delta_{ij} \xi_{ij}$ for $i \leq j$. $\delta_{ij}$ are i.i.d. Bernoulli random variables taking the value $1$ with probability $p \geq n^{-1+\delta}$ for any constant $\delta > 0$ and $\xi_{ij}$ are i.i.d. centered, subgaussian random variables. We show that with high probability this class of random matrices has simple spectrum (i.e. the eigenvalues appear with multiplicity one). We can slightly modify our proof to show that the adjacency matrix of a sparse Erd\H{o}s-R\'enyi graph has simple spectrum for $n^{-1+\delta } \leq p \leq 1- n^{-1+\delta}$. These results are optimal in the exponent. The result for graphs has connections to the notorious graph isomorphism problem.