We describe the class of probability measures whose moments are given in terms of the Aval numbers. They are expressed as the multiplicative free convolution of measures corresponding to the ballot numbers .
In [AnsMonic, AnsBoolean], we investigated monic multivariate non-commutative orthogonal polynomials, their recursions, states of orthogonality, and corresponding continued fraction expansions. In this note, we collect a number of examples, demonstrating what these general results look like for the most important states on non-commutative polynomials, namely for various product states. In particular, we introduce a notion of a product-type state on polynomials, which covers all the non-commutative...
This is a presentation of recent work on quantum permutation groups. Contains: a short introduction to operator algebras and Hopf algebras; quantum permutation groups, and their basic properties; diagrams, integration formulae, asymptotic laws, matrix models; the hyperoctahedral quantum group, free wreath products, quantum automorphism groups of finite graphs, graphs having no quantum symmetry; complex Hadamard matrices, cocycle twists of the symmetric group, quantum groups acting on 4 points; remarks...
In this article we discuss the Catalan and super-Catalan (or Schröder) numbers. We start with some combinatorial interpretations of those numbers. We study two probability measures in the context of free probability, one whose moments are super-Catalan, and another, whose even moments are super-Catalan and odd moments are zero. With the use of the latter we also show some new formulae for evaluation of the Catalans in terms of super-Catalans and vice-versa.
We will show that the conditional first moment of the free deformed Poisson random variables (q = 0) corresponding to operators fulfilling the free relation is a linear function of the regression and the conditional variance also is a linear function of the regression. For this purpose we will first demonstrate some properties of the Wick product and then we will concentrate on the free deformed Poisson random variables.
Belinschi and Nica introduced a composition semigroup of maps on the set of probability measures. Using this semigroup, they introduced a free divisibility indicator, from which one can know quantitatively if a measure is freely infinitely divisible or not. In the first half of the paper, we further investigate this indicator: we calculate how the indicator changes with respect to free and Boolean powers; we prove that free and Boolean 1/2-stable laws have free divisibility indicators equal to infinity;...
Given an orthogonal projection P and a free unitary Brownian motion in a W*-non commutative probability space such that Y and P are *-free in Voiculescu’s sense, we study the spectral distribution νₜ of Jₜ = PYₜPYₜ*P in the compressed space. To this end, we focus on the spectral distribution μₜ of the unitary operator SYₜSYₜ*, S = 2P - 1, whose moments are related to those of Jₜ via a binomial-type expansion already obtained by Demni et al. [Indiana Univ. Math. J. 61 (2012)]. In this connection,...