  The current work applies some recent combinatorial tools due to Jain to
control the eigenvalue gaps of a matrix $M_n = M + N_n$ where $M$ is
deterministic, symmetric with large operator norm and $N_n$ is a random
symmetric matrix with subgaussian entries. One consequence of our tail bounds
is 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 of
both the probability and the size of the matrix $M$.
