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).

Abstract. In this paper we characterize the possible outliers in the spectrum of large deformed unitarily invariant additive and multiplicative models, as well as the eigenvectors corresponding to them. We allow both the non-deformed unitarily invariant model and the perturbation matrix to have non-trivial limiting spectral measures and spiked outliers in their spectrum. We uncover a remarkable new phenomenon: a single spike can generate asymptotically several outliers in the spectrum of the deformed model. The free subordination functions play a key role in this analysis.

We consider the framework of an operator-valued noncommutative probability space over a unital C*-algebra B. We show how for a B-valued distribution \mu one can define convolution powers with respect to free additive convolution and with respect to Boolean convolution, where the exponent considered in the power is a suitably chosen linear map \eta from B to B, instead of being a non-negative real number. More precisely, the Boolean convolution power is defined whenever \eta is completely positive, while the free additive convolution power is defined whenever \eta - 1 is completely positive (where 1 stands for the identity map on B). In connection to these convolution powers we define an evolution semigroup related to the Boolean Bercovici-Pata bijection. We prove several properties of this semigroup, including its connection to the B-valued free Brownian motion. We also obtain two results on the operator-valued analytic function theory related to the free additive convolution powers with exponent \eta. One of the results concerns analytic subordination for B-valued Cauchy-Stieltjes transforms. The other gives a B-valued version of the inviscid Burgers equation, which is satisfied by the Cauchy-Stieltjes transform of a B-valued free Brownian motion.Comment: 33 pages, no figure

Free probabilistic considerations of type B first appeared in the paper of Biane, Goodman and Nica [P. Biane, F. Goodman, A. Nica, Non-crossing cumulants of type B, Trans. Amer. Math. Soc. 355 (2003Soc. 355 ( ) 2263Soc. 355 ( -2303. Recently, connections between type B and infinitesimal free probability were put into evidence by Belinschi and Shlyakhtenko [S.T. Belinschi, D. Shlyakhtenko, Free probability of type B: Analytic aspects and applications, preprint, 2009, available online at www.arxiv.org under reference arXiv:0903.2721]. The interplay between "type B" and "infinitesimal" is also the object of the present paper. We study infinitesimal freeness for a family of unital subalgebras A 1 , . . . , A k in an infinitesimal noncommutative probability space (A, ϕ, ϕ ) and we introduce a concept of infinitesimal non-crossing cumulant functionals for (A, ϕ, ϕ ), obtained by taking a formal derivative in the formula for usual non-crossing cumulants. We prove that the infinitesimal freeness of A 1 , . . . , A k is equivalent to a vanishing condition for mixed cumulants; this gives the infinitesimal counterpart for a theorem of Speicher from "usual" free probability. We show that the lattices NC (B) (n) of non-crossing partitions of type B appear in the combinatorial study of (A, ϕ, ϕ ), in the formulas for infinitesimal cumulants and when describing alternating products of infinitesimally free random variables. As an application of alternating free products, we observe the infinitesimal analogue for the well-known fact that freeness is preserved under compression with a free projection. As another application, we observe the infinitesimal analogue for a well-known procedure used to construct free families of free Poisson elements. Finally, we discuss situations when the freeness of A 1 , . . . , A k in (A, ϕ)

We study how Boolean cumulants can be used in order to address operations with freely independent random variables, particularly in connection to the * -distribution of the product of two selfadjoint freely independent random variables, and in connection to the distribution of the anticommutator of such random variables.