A large deviation principle for Wigner matrices without Gaussian tails (Q471151)

Let \(H\) be an \(n \times n\) square Hermitian matrix with i.i.d.\ entries with tail \(P(|H_{ij}|>t) \leq \exp(-a t^b)\) with \(a>0\) and \(1<b<2\). The main result is a large deviations principle for the empirical spectral distribution of \(H/\sqrt{n}\) with speed \(n^{1+b/2}\) and good rate function related to the free divisibility with respect to the semicircle law. The proof is based on a clever additive decomposition of the matrix. This is to date the unique example of a large deviations principle for the empirical spectral distribution of a nonunitary invariant ensemble of random matrices.