By Azzouz Dermoune, Philippe Heinrich (auth.), Catherine Donati-Martin, Michel Émery, Alain Rouault, Christophe Stricker (eds.)

ISBN-10: 3540779124

ISBN-13: 9783540779124

Stochastic methods are as ordinary the most topic of the Séminaire, with contributions on Brownian movement (fractional or other), Lévy approaches, martingales and probabilistic finance. different probabilistic subject matters also are current: huge random matrices, statistical mechanics. The contributions during this quantity supply a sampling of contemporary effects on those themes. All contributions apart from are written in English language.

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**Extra info for Séminaire de Probabilités XLI**

**Sample text**

2. Indeed, under the assumptions made on J, one readily derives that 1 log PN [x∗N ≤ t] = −∞, ∀t < bV lim N N whereas 1 log PN [x∗N ≥ t] = −βJ(t), ∀t > bV . 3. Next, we turn to the proof of the main result of this section. 2: It remains to treat the case where t ≥ 1. : Let us first show that ∀t > bV , L(t) > 0 (23) with L(t) given by L(t) = FVλ,t − FVλ . In fact, it suffices to observe the following identity Mλ,t := Mλ ∩ M([0, t]) = {μ ∈ M(R+ ) / 0 ≤ μ ≤ λ/[0,t] } which means that Mλ,t is the class of probability measures submitted to the constraint λ/[0,t] .

It appears that the limiting values of the moments (N,b) M2k =E 1 Tr H (N,b) N 2k crucially depend of the ratio between b and N when N → ∞ (see [5, 13, 17]). (N,b) If b/N → 1 as N → ∞, then M2k → mk and the semicircle law is valid (N,b) in this case. If b/N → c and 0 < c < 1, then the limiting values of M2k differ from mk . Finally, if 1 ≪ b ≪ N , then the semicircle law is valid again. The last asymptotic regime of (relatively) narrow band width attracts a special interest from researchers. In this case the spectral properties of band random matrices exhibit a transition from one type to another.

Immediate applications include the Gaussian Orthogonal Ensemble and the ensemble of Gaussian anti-symmetric Hermitian matrices. Finally we apply our method to the ensemble of N × N Gaussian Hermitian random matrices H (N,b) whose elements are zero outside the band of width b. The other elements are taken from GUE; the matrix obtained is renormalized by b−1/2 . We derive estimates for the moments of H (N,b) and prove that the spectral norm H (N,b) remains bounded in the limit N, b → ∞ when (log N )3/2 /b → 0.

### Séminaire de Probabilités XLI by Azzouz Dermoune, Philippe Heinrich (auth.), Catherine Donati-Martin, Michel Émery, Alain Rouault, Christophe Stricker (eds.)

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