Tail bounds for gaps between eigenvalues of sparse random matrices

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 work of the second author and Vu. We also show that for sparse Erdős–Rényi graphs, weak and strong nodal domains are the same, answering a question of Dekel, Lee, and Linial.