Serban T. Belinschi

Free Bessel Laws

We introduce and study a remarkable family of real probability measures that we call free Bessel laws. These are related to the free Poisson law. 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.

The normal distribution is -infinitely divisible

Abstract 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 sub-family of Askey–Wimp–Kerov distributions are freely infinitely divisible, of which the normal distribution is a special case. At the time of this writing this is only the third example known to us of a nontrivial distribution that is infinitely divisible with respect to both classical and free convolution, the others being the Cauchy distribution and the free 1/2-stable distribution.

Outliers in the spectrum of large deformed unitarily invariant models

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.

Eigenvectors and eigenvalues in a random subspace of a tensor product

Given two positive integers n and k and a parameter t ∈ (0, 1), we choose at random a vector subspace Vn ⊂ C k ⊗ C n of dimension N ∼ tnk. We show that the set of k-tuples of singular values of all unit vectors in Vn fills asymptotically (as n tends to infinity) a deterministic convex set Kk,t that we describe using a new norm in R . Our proof relies on free probability, random matrix theory, complex analysis and matrix analysis techniques. The main result result comes together with a law of large numbers for the singular value decomposition of the eigenvectors corresponding to large eigenvalues of a random truncation of a matrix with high eigenvalue degeneracy.

FREE BROWNIAN MOTION AND EVOLUTION TOWARDS ⊞-INFINITE DIVISIBILITY FOR k-TUPLES

Let D be the space of non-commutative distributions of k-tuples of selfadjoints in a C*-probability space (for a fixed k). We introduce a semigroup of transformations B_t of D, such that every distribution in D evolves under the B_t towards infinite divisibility with respect to free additive convolution. The very good properties of B_t come from some special connections that we put into evidence between free additive convolution and the operation of Boolean convolution. On the other hand we put into evidence a relation between the transformations B_t and free Brownian motion. More precisely, we introduce a transformation Phi of D which converts the free Brownian motion started at an arbitrary distribution m in D into the process B_t (Phi(m)), t>0.

Convolution powers in the operator-valued framework

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.

Squared eigenvalue condition numbers and eigenvector correlations from the single ring theorem

We extend the so-called single ring theorem[1], also known as the Haagerup-Larsen theorem[2], by showing that in the limit when the size of the matrix goes to infinity a particular correlator between left and right eigenvectors of the relevant non-hermitian matrix

A NONCOMMUTATIVE VERSION OF THE JULIA-WOLFF-CARATHEODORY THEOREM

X

Some Geometric Properties of the Subordination Function Associated to an Operator-Valued Free Convolution Semigroup

, being the spectral density weighted by the squared eigenvalue condition number, is given by a simple formula involving only the radial spectral cumulative distribution function of

On a remarkable semigroup of homomorphisms with respect to free multiplicative convolution

X