# Tail bounds for gaps between eigenvalues of sparse random matrices

## TL;DR

This paper establishes the first eigenvalue repulsion bounds for sparse random matrices, demonstrating simple spectra and applying these results to Erdős–Rényi graphs to unify weak and strong nodal domains.

## Contribution

It introduces novel eigenvalue gap bounds for sparse matrices, extending previous work and improving sparsity and error probability ranges.

## Key findings

- Sparse matrices have simple spectra due to eigenvalue repulsion.
- Eigenvalue tail bounds are established for sparse random matrices.
- Weak and strong nodal domains coincide in sparse Erdős–Rényi graphs.

## Abstract

We prove the first eigenvalue repulsion bound for sparse random matrices. As a consequence, we show that these matrices have simple spectrum, improving the range of sparsity and error probability from the work of the second author and Vu. As an application of our tail bounds, we show that for sparse Erd\H{o}s--R\'enyi graphs, weak and strong nodal domains are the same, answering a question of Dekel, Lee, and Linial.