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Large Deviations of the Maximum Eigenvalue

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

Abstract

We here restrict ourselves to the case where V (x) = ?x 2/4 and for short denote by \(P_\beta ^N\) the law of the eigenvalues (?i)1?i?N:

$$P_\beta ^N (d\lambda _1 , \cdots ,d\lambda _N ) = \frac{1}{{Z_\beta ^N }}\prod\limits_{1 \le i < j \le N} {|\lambda _i - \lambda _j |^\beta } \prod\limits_{1 \le i \le N} {e^{ - \frac{{\beta N\lambda _i^2 }}{4}} d\lambda _i }$$

with

$$Z_\beta ^N = \int {\prod\limits_{1 \le i < j \le N} {|\lambda _i - \lambda _j |^\beta } \prod\limits_{1 \le i \le N} {e^{ - \frac{{\beta N\lambda _i^2 }}{4}} d\lambda _i } } .$$

Selberg (cf. [153, Theorem 4.1.1] or [6]) found the explicit formula for \(Z_\beta ^N\) for any ? ? 0:

$$Z_\beta ^N = (2\pi )^{\frac{N}{2}} \left( {\frac{{\beta N}}{2}} \right)^{ - \beta N(N - 1)/4 - \frac{N}{2}} \prod\limits_{J = 1}^N {\frac{{\Gamma \left( {\frac{{j\beta }}{2}} \right)}}{{\Gamma \left( {\frac{\beta }{2}} \right)}}} .$$
((11.1))

The knowledge of \(Z_\beta ^N\) up to the second order is crucial below, reason why we restrict ourselves to quadratic potentials in the next theorem (see Exercise 11.4 for a slight extension).

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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 Deviations of the Maximum Eigenvalue. 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_12

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