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The current work applies some recent combinatorial tools due to Jain tocontrol the eigenvalue gaps of a matrix $M_n = M + N_n$ where $M$ isdeterministic, symmetric with large operator norm and $N_n$ is a randomsymmetric matrix with subgaussian entries. One consequence of our tail boundsis that $M_n$ has simple spectrum with probability at least $1 -\exp(-n^{2/15})$ which improves on a result of Nguyen, Tao and Vu in terms ofboth the probability and the size of the matrix $M$.

Eigenvalue Gaps of Random Perturbations of Large Matrices