Eigenvalue Gaps of Random Perturbations of Large Matrices

Abstract

The current work applies some recent combinatorial tools due to Jain to control the eigenvalue gaps of a matrix Mn = M + Nn where M is deterministic, symmetric with large operator norm and Nn is a random symmetric matrix with subgaussian entries. One consequence of our tail bounds is that Mn has simple spectrum with probability at least 1 − exp(−n 2/15 ) which improves on a result of Nguyen, Tao and Vu in terms of both the probability and the size of the matrix M .