Random matrices have simple spectrum

Let $M_n = (ξ_{ij})_{1 \leq i,j \leq n}$ be a real symmetric random matrix in which the upper-triangular entries $ξ_{ij}, i<j$ and diagonal entries $ξ_{ii}$ 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.

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