A LARGE DEVIATION PRINCIPLE FOR WIGNER MATRICES WITHOUT GAUSSIAN TAILS

We consider n x n Hermitian matrices with i.i.d. entries X-ij whose tail probabilities P(vertical bar X-ij vertical bar >= t) behave like e(-at alpha) for some a > 0 and alpha is an element of (0, 2). We establish a large deviation principle for the empirical spectral measure of X/root n with speed n(1+alpha/2) with a good rate function J(mu) that is finite only if mu is of the form mu = mu(sc) boxed plus nu for some probability measure nu on R, where boxed plus denotes the free convolution and mu(sc) is Wigner's semicircle law. We obtain explicit expressions for J(mu(sc) boxed plus nu) in terms of the alpha th moment of nu. The proof is based on the analysis of large deviations for the empirical distribution of very sparse random rooted networks.