Large deviation principle for the law of the spectral measure of shifted Wigner matrices

Abstract

The goal of this section is to prove the following theorem.

Theorem 13.1. Assume that D_N is uniformly bounded with spectral measure converging to ?_D. Let X^?,N be a Gaussian symmetric (resp. Hermitian) Wigner matrix when ? = 1 (resp. ? = 2). Then the law of the spectral measure L_{Y N,?} of the Wigner matrix Y^N,? = D_N + X^?,N satisfies a large deviation principle in the scale N² with a certain good rate function J?(?D, .).

We shall base our approach on Bryc’s theorem, that says that the above large deviation principle statement is equivalent to the fact that for any bounded continuous function f on P(R),

$$\Lambda (f) = \mathop {\lim }\limits_{N \to \infty } \frac{1}{{N^2 }}\log \int {e^{N^2 f(L_Y N,\beta )} dP}$$

exists and is given by $- \inf \{ J_\beta (\mu D,v) - f(v)\}$. It is not clear how one could a priori study such limits, except for very trivial functions f. However, if we consider the matrix-valued process $Y^{N,\beta } (t) = D^N + H^{N,\beta } (t)$ with Brownian motion H^N,? described in (V.1) and its spectral measure process

$$L_{N,\beta } (t): = L_Y N,\beta _{(t)} = \frac{1}{N}\sum\limits_{i = 1}^N \delta \lambda _i (Y^{N,\beta } (t)) \in P(R),$$

we may construct martingales by use of Itô’s calculus. Indeed, continuous martingales lead to exponential martingales, which have constant expectation, and therefore allow one to compute the exponential moments of a whole family of functionals of L_N(t). This idea gives easily a large deviation upper bound for the law of (L_N,?(t), t ? [0, 1]), and therefore for the law of L_{Y N,?}, that is, the law of L_N,?(1). The difficult point here is to check that this bound is sharp, i.e., it is enough to compute the exponential moments of this family of functionals in order to obtain the large deviation lower bound.