Let $M_n = (\xi_{ ij})_{ 1 \leq i,j \leq n}$ be a real symmetric random matrix in which the upper-triangular entries $\xi_{ ij}, i<j$ and diagonal entries $\xi_{ ii}$ are independent. We show that with probability tending to 1, $M_n$ has no repeated eigenvalues. As a corollary, we deduce that the Erd{ \H o}s-Renyi random graph has simple spectrum asymptotically almost surely, answering a question of Babai.
Categories: math.PR math.CO