Abstract. We introduce and study a remarkable family of real probability measures πst that we call free Bessel laws. These are related to the free Poisson law π via the formulae π s1 = π ⊠s and π 1t = π ⊞t . Our study includes definition and basic properties, analytic aspects (supports, atoms, densities), combinatorial aspects (functional transforms, moments, partitions), and a discussion of the relation with random matrices and quantum groups.

In this paper, we investigate the asymptotic spectrum of complex or real Deformed Wigner matrices (MN )N defined by MN = WN / √ N + AN where WN is an N × N Hermitian (resp., symmetric) Wigner matrix whose entries have a symmetric law satisfying a Poincaré inequality. The matrix AN is Hermitian (resp., symmetric) and deterministic with all but finitely many eigenvalues equal to zero. We first show that, as soon as the first largest or last smallest eigenvalues of AN are sufficiently far from zero, the corresponding eigenvalues of MN almost surely exit the limiting semicircle compact support as the size N becomes large. The corresponding limits are universal in the sense that they only involve the variance of the entries of WN . On the other hand, when AN is diagonal with a sole simple nonnull eigenvalue large enough, we prove that the fluctuations of the largest eigenvalue are not universal and vary with the particular distribution of the entries of WN .

We investigate the asymptotic behavior of the eigenvalues of spiked perturbations of Wigner matrices defined by MN = 1 √ N WN + AN , where WN is a N × N Wigner Hermitian matrix whose entries have a distribution µ which is symmetric and satisfies a Poincaré inequality and AN is a deterministic Hermitian matrix whose spectral measure converges to some probability measure ν with compact support. We assume that AN has a fixed number of fixed eigenvalues (spikes) outside the support of ν whereas the distance between the other eigenvalues and the support of ν uniformly goes to zero as N goes to infinity. We establish that only a particular subset of the spikes will generate some eigenvalues of MN which will converge to some limiting points outside the support of the limiting spectral measure. This phenomenon can be fully described in terms of free probability involving the subordination function related to the free additive convolution of ν by a semicircular distribution. Note that only finite rank perturbations had been considered up to now (even in the deformed GUE case).

We show how the Clark-Ocone-Haussmann formula for Brownian motion on a compact Riemannian manifold put forward by S. Fang in his proof of the spectral gap inequality for the Ornstein-Uhlenbeck operator on the path space can yield in a very simple way the logarithmic Sobolev inequality on the same space. By an appropriate integration by parts formula the proof also yields in the same way a logarithmic Sobolev inequality for the path space equipped with a general diffusion measure as long as the torsion of the corresponding Riemannian connection satisfies Driver's total antisymmetry condition. IHwhere D : L 2 (µ) → L 2 (µ; IH) is the (Malliavin) gradient operator on W 0 (IR n ) and IH the Cameron-Martin Hilbert space. Logarithmic Sobolev inequalities were introduced to study

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In this paper, we study the fluctuations of the extreme eigenvalues of a spiked finite rank deformation of a Hermitian (resp. symmetric) Wigner matrix when these eigenvalues separate from the bulk. We exhibit quite general situations that will give rise to universality or non universality of the fluctuations, according to the delocalization or localization of the eigenvectors of the perturbation. Dealing with the particular case of a spike with multiplicity one, we also establish a necessary and sufficient condition on the associated normalized eigenvector so that the fluctuations of the corresponding eigenvalue of the deformed model are universal. RésuméDans ce papier, nousétudions les fluctuations des valeurs propres extrémales d'une matrice de Wigner hermitienne (resp. symétrique) déformée par une perturbation de rang fini dont les valeurs propres non nulles sont fixées, dans le cas où ces valeurs propres extrémales se détachent du reste du spectre. Nous décrivons des situations générales d'universalité ou de non-universalité des fluctuations correspondant au caractère localisé ou délocalisé des vecteurs propres de la perturbation. Lorsque l'une des valeurs propres de la perturbation est de multiplicité un, nousétablissons de plus une condition nécessaire et suffisante sur le vecteur propre associé pour que les fluctuations de la valeur propre correspondante du modèle déformé soient universelles.Mathematics Subject Classification (2010): 60B20, 15A18, 60F05 .

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