A large deviation principle for Wigner matrices without Gaussian tails. (English) Zbl 1330.60012
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.
Reviewer: Djalil Chafaï (Paris)
MSC:
60B20 | Random matrices (probabilistic aspects) |
15B52 | Random matrices (algebraic aspects) |
47A10 | Spectrum, resolvent |
15A18 | Eigenvalues, singular values, and eigenvectors |