K-divisible partitions in free probability

Abstract

In this thesis we study the role of k-divisible non-crossing partitions in Free Probability. First, we consider the combinatorial convolution ∗ in the lattices NC of non-crossing partitions and NC of k-divisible non-crossing partitions. We show that convolving k times with the zeta-function in NC is equivalent to convolving once with the zeta-function in NC. This gives new ways of counting objects like k-equal partitions, k-divisible partitions and k-multichains both in NC and NC. We also consider some statistics of block sizes in k-divisible non-crossing partitions. Second, we introduce and study the notion of k-divisible elements in a non-commutative probability space. A k-divisible element is a (non-commutative) random variable whose n-th moment vanishes whenever n is not a multiple of k. For such k-divisible element x, we derive a formula for the free cumulants of x in terms of the free cumulants of x. For this we use our combinatorial results on the lattice of k-divisible non-crossing partitions. We prove that if a and s are free and s is k-divisible then sps and a are free, where p is any polynomial (in a and s) of degree k − 2 in s. Moreover, we define a notion of R-diagonal k-tuples and prove similar results. Next, we show that free multiplicative convolution between a measure concentrated on the positive real line and a probability measure with k-symmetry is well defined. Analytic tools to calculate this convolution are developed. We then concentrate on free additive powers of k-symmetric distributions and prove that μ t is a well defined probability measure, for all t > 1. We derive central limit theorems and Poisson type ones. More generally, we consider freely infinitely divisible measures and prove that free infinite divisibility is maintained under the mapping μ→ μ. Relations between free multiplicative powers and k-divisible non-crossing partitions are also found and generalized to any product of free random variables. We conclude by focusing on (k-symmetric) free stable distributions, for which we prove a reproducing property generalizing the ones known for one sided and real symmetric free stable laws.