Smoothed analysis of symmetric random matrices with continuous distributions

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 $\mathrm{\mid }(D+R{)}^{-1}\mathrm{\mid }=O(n^2)$ 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.