Large deviations for the law of the spectral measure of Gaussian Wigner's matrices

Abstract

In this section, we consider the law of N random variables (?₁, . . . , ?_N) with

$$P_{V,\beta }^N (d\lambda _1 , \ldots ,d\lambda _N ) = (Z_{V,\beta }^N )^{ - 1} |\Delta (\lambda )|^\beta e - N\sum\nolimits_{i = 1}^N {V(\lambda _i )\prod\limits_{i = 1}^N {d\lambda _i } ,} $$

((10.1))

for a continuous function V : R ? R such that

$$\mathop {\lim {\rm inf}}\limits_{|x| \to \infty } \frac{{V(x)}}{{\beta \log |x|}} > 1$$

((10.2))

and a positive real number ?. Here, ?(?) = ?_1?i<j?N(?_i ? ?_j ).

When V (x) = 4^-1?x², we have seen in Lemma IV that $P_{4^{ - 1} \beta x^2 ,\beta }^N$ is the law of the eigenvalues of an N ×N GOE (resp. GUE, resp GSE) matrix when ? = 1 (resp. ? = 2, resp. ? = 4). The case ? = 4 corresponds to another matrix ensemble, namely the GSE. In view of these remarks and other applications discussed in Part III, we consider in this section the slightly more general model with a potential V . We emphasize, however, that the distribution (10.1) precludes us from considering random matrices with independent non-Gaussian entries.