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

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Part of the Lecture Notes in Mathematics book series (LNMECOLE,volume 1957)

Abstract

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

Theorem 13.1. Assume that DN 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 LY N,? of the Wigner matrix YN,? = DN + 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 HN,? 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 LN(t). This idea gives easily a large deviation upper bound for the law of (LN,?(t), t ? [0, 1]), and therefore for the law of LY N,?, that is, the law of LN,?(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.

Keywords

  • Brownian Motion
  • Spectral Measure
  • Large Deviation Principle
  • Fourth Moment
  • Hydrodynamical Limit

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Correspondence to Alice Guionnet .

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© 2009 Springer-Verlag Berlin Heidelberg

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Guionnet, A. (2009). Large deviation principle for the law of the spectral measure of shifted Wigner matrices. In: Large Random Matrices: Lectures on Macroscopic Asymptotics. Lecture Notes in Mathematics(), vol 1957. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-69897-5_14

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