A large deviation principle for Wigner matrices without Gaussian tails

We consider n× n Hermitian matrices with i.i.d. entries Xᵢⱼ whose tail probabilities P (|Xᵢⱼ|≥ t) behave like e^-atα for some $a>0$ and α ∈(0,2). We establish a large deviation principle for the empirical spectral measure of X/√n with speed n1+α /2 with a good rate function J(μ) that is finite only if μ is of the form μ=μscν for some probability measure ν on R, where denotes the free convolution and μsc is Wigner’s semicircle law. We obtain explicit expressions for J(μscν) in terms of the αth moment of ν. The proof is based on the analysis of large deviations for the empirical distribution of very sparse random rooted networks.