  We study invertibility of matrices of the form $D+R$ where $D$ is an
arbitrary symmetric deterministic matrix, and $R$ is a symmetric random matrix
whose independent entries have continuous distributions with bounded densities.
We show that $|(D+R)^{-1}| = O(n^2)$ with high probability. The bound is
completely independent of $D$. No moment assumptions are placed on $R$; in
particular the entries of $R$ can be arbitrarily heavy-tailed.
