Eigenvalue gaps of random perturbations of large matrices

This work applies some recent combinatorial tools due to Jain to control the eigenvalue gaps of a matrix [Formula: see text] where [Formula: see text] is deterministic, symmetric with large operator norm and [Formula: see text] is a random symmetric matrix with sub-Gaussian entries. One consequence of our tail bounds is that [Formula: see text] has simple spectrum with probability at least [Formula: see text] which improves on a result of Nguyen, Tao and Vu in terms of both the probability and the size of the matrix [Formula: see text].