| Résume | Let $W$ be an $n\times n$ symmetric matrix with i.i.d centered entries and unit variance. The celebrated Wigner's theorem states that the empirical law of eigenvalues of $W/\sqrt{n}$ converges weakly to the semicircle law, a measure supported in $[-2,2]$. For the largest eigenvalue, Bai and Yin showed that it converges to $2$ if and only if the entries of $W$ have a bounded fourth moment, namely, $W$ does not have outliers. In this talk, we explore the universality and stability of Bai-Yin's result under sparsification. In other words, we consider the random matrix $X=\Sigma \circ W$, where $\Sigma$ is a deterministic matrix, and $\circ$ denotes the Hadamard product. We contribute sharp conditions for subgaussian matrices $X$ to have outliers in terms of non-structural parameters of $\Sigma$.
This is a joint work with Dylan Altschuler, Konstantin Tikhomirov, and Pierre Youssef.
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