Alexandru Nica

Annular noncrossing permutations and partitions, and second-order asymptotics for random matrices

We study the set Sann−nc(p,q) of permutations of {1, …, p+q} which are noncrossing in an annulus with p points marked on its external circle and q points marked on its internal circle. We define Sann−nc(p,q,q) algebraically by identifying the crossing patterns which can occur in an annulus. We prove the annular counterpart for a “geodesic condition” shown by Biane to characterize noncrossing permutations in a disc. We examine the relation between Sann−nc(p,q,q) and the set NC ann (p,q of annular noncrossing partitions of {1, …, p+q} and observe that (unlike in the disc case) the natural map from Sann−nc(p,q) onto NC ann (p,q) has a pathology which prevents it from being injective. We point out that annular noncrossing permutations appear in the description of the second-order asymptotics for the joint moments of certain families (Wishart and GUE) of random matrices. Some of the formulas extend to a multiannular framework; as an application of that, we observe a phenomenon of asymptotic Gaussianity for traces of words made with independent Wishart matrices.

Commutators of free random variables

Let A be a unital C∗-algebra, given together with a specified state φ : A → C. Consider two selfadjoint elements a, b of A, which are free with respect to φ (in the sense of the free probability theory of Voiculescu). Let us denote c := i(ab− ba), where the i in front of the commutator is introduced to make c selfadjoint. In this paper we show how the spectral distribution of c can be calculated from the spectral distributions of a and b. Some properties of the corresponding operation on probability measures are also discussed. The methods we use are combinatorial, based on the description of freeness in terms of non-crossing partitions; an important ingredient is the notion of R-diagonal pair, introduced and studied in our previous paper [12].

A “Fourier Transform” for Multiplicative Functionson Non-Crossing Partitions

We describe the structure of the group of normalized multiplicative functions on lattices of non-crossing partitions. As an application, we give a combinatorial proof of a theorem of D. Voiculescu concerning the multiplication of free random variables

A one-parameter family of transforms, linearizing convolution laws for probability distributions

We study a family of transforms, depending on a parameterq∈[0,1], which interpolate (in an algebraic framework) between a relative (namely: -iz(log ℱ(·))′(-iz)) of the logarithm of the Fourier transform for probability distributions, and its free analogue constructed by D. Voiculescu ([16, 17]). The classical case corresponds toq=1, and the free one toq=0.We describe these interpolated transforms: (a) in terms of partitions of finite sets, and their crossings; (b) in terms of weighted shifts; (c) by a matrix equation related to the method of Stieltjes for expanding continuedJ-fractions as power series. The main result of the paper is that all these descriptions, which extend basic approaches used forq=0 and/orq=1, remain equivalent for arbitraryq∈[0, 1].We discuss a couple of basic properties of the convolution laws (for probability distributions) which are linearized by the considered family of transforms (these convolution laws interpolate between the usual convolution — atq=1, and the free convolution introduced by Voiculescu — atq=0). In particular, we note that description (c) mentioned in the preceding paragraph gives an insight of why the central limit law for the interpolated convolution has to do with theq-continuous Hermite orthogonal polynomials.

Operator-valued distributions. I. Characterizations of freeness

Let

R-Diagonal Elements and Freeness With Amalgamation

M

ON A GROUPOID CONSTRUCTION FOR ACTIONS OF CERTAIN INVERSE SEMIGROUPS

be a

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

B

On the number of cycles of given length of a free word in several random permutations

-probability space. Assume that

On the Distribution of the Area Enclosed by a Random Walk onZ2

B