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.

Abstract. We develop an analytic theory of operator-valued additive free convolution in terms of subordination functions. In contrast to earlier investigations our functions are not just given by power series expansions, but are defined as Frechet analytic functions in all of the operator upper half plane. Furthermore, we do not have to assume that our state is tracial. Combining this new analytic theory of operator-valued free convolution with Anderson's selfadjoint version of the linearization trick we are able to provide a solution to the following general random matrix problem: Let X

Let M denote the space of Borel probability measures on R. For every t ≥ 0 we consider the transformation B t : M → M defined bywhere ⊞ and ⊎ are the operations of free additive convolution and respectively of Boolean convolution on M, and where the convolution powers with respect to ⊞ and ⊎ are defined in the natural way. We show that B s • B t = B s+t , ∀ s, t ≥ 0 and that, quite surprisingly, every B t is a homomorphism for the operation of free multiplicativeWe prove that for t = 1 the transformation B 1 coincides with the canonical bijection B : M → M inf−div discovered by Bercovici and Pata in their study of the relations between infinite divisibility in free and in Boolean probability. Here M inf−div stands for the set of probability distributions in M which are infinitely divisible with respect to the operation ⊞. As a consequence, we have that B t (µ) is ⊞-infinitely divisible for every µ ∈ M and every t ≥ 1.On the other hand we put into evidence a relation between the transformations B t and the free Brownian motion; indeed, Theorem 4 of the paper gives an interpretation of the transformations B t as a way of re-casting the free Brownian motion, where the resulting process becomes multiplicative with respect to ⊠, and always reaches ⊞-infinite divisibility by the time t = 1.

Let D c (k) be the space of (non-commutative) distributions of k-tuples of selfadjoint elements in a C * -probability space. On D c (k) one has an operation of free additive convolution, and one can consider the subspace D inf-div c (k) of distributions which are infinitely divisible with respect to this operation. The linearizing transform for is the R-transform (one has R μ ν = R μ + R ν , ∀μ, ν ∈ D c (k)). We prove that, with M μ the moment series of μ. (The series η μ is the counterpart of R μ in the theory of Boolean convolution.) As a consequence, one can define a bijection B :We show that B is a multi-variable analogue of a bijection studied by Bercovici and Pata for k = 1, and we prove a theorem about convergence in moments which parallels the Bercovici-Pata result. On the other hand we prove the formulawith μ, ν considered in a space D alg (k) ⊇ D c (k) where the operation of free multiplicative convolution always makes sense. An equivalent reformulation of (II) is thatwhere is an operation on series previously studied by Nica and Speicher, and which describes the multiplication of free k-tuples in terms of their R-transforms. Formula (III) shows that, in a certain sense, η-series behave in the same way as R-transforms in connection to the operation of multiplication of free k-tuples of non-commutative random variables.

We show that the subordination results of D. Voiculescu and Ph. Biane can be deduced from a continuity property of fixed points for analytic functions.

Consider a Borel probability measure µ on the real line, and denote by {µ t : t ≥ 1} the free additive convolution semigroup defined by Nica and Speicher. We show that the singular part of µ t is purely atomic and the density of µ t is locally analytic, provided that t > 1. The main ingredient is a global inversion theorem for analytic functions on a half plane. (2000): 46L54, 30A99 Mathematics Subject Classification

We prove that the classical normal distribution is infinitely divisible with respect to the free additive convolution. We study the Voiculescu transform first by giving a survey of its combinatorial implications and then analytically, including a proof of free infinite divisibility. In fact we prove that a subfamily Askey-Wimp-Kerov distributions are freely infinitely divisible, of which the normal distribution is a special case.The Cauchy-Stieltjes transform of a measure µ on the real line is defined bywhen µ is a positive measure, this function maps C + into the lower half-plane. For details we refer to the excellent book of Akhieser [1]. For c = 0, we have µ 0 = γ, the normal distribution [27], and one can easily check that one can extend this family continuously to c = −1 by letting µ −1 = δ 0 , the probability giving mass one to {0}. This family, introduced in [4], plays an important role in [27]; we shall call its members the Askey-Wimp-Kerov distributions. It will turn out from our proof that {µ c : c ∈ [−1, 0]} are freely infinitely divisible. Numerical computations show that for several values of c > 0, µ c is not freely infinitely divisible. Numerical evidence seems also to indicate that µ c is classicaly infinitely divisible only when c = 0 or c = −1. An interesting interpolation between the normal and the semicircle law was constructed by Bryc, Dembo and Jiang [18] and further investigated by Buchholz [19]. This leads to a generalized Brownian motion, given by a weight function (0 < b < 1)

Abstract. We study the asymptotic behavior of the appropriately scaled and possibly perturbed spectral measureμ of large random real symmetric matrices with heavy tailed entries. Specifically, consider theis an infinite array of i.i.d real variables with common distribution in the domain of attraction of an α-stable law, α ∈ (0, 2), and σ is a deterministic function. For random diagonal D N independent of Y σ N and with appropriate rescaling a N , we prove thatμ aconverges in mean towards a limiting probability measure which we characterize. As a special case, we derive and analyze the almost sure limiting spectral density for empirical covariance matrices with heavy tailed entries.