Products of free random variables and $k$-divisible non-crossing partitions

Abstract

We derive a formula for the moments and the free cumulants of the multiplication of k free random variables in terms of k-equal and k-divisible non-crossing partitions. This leads to a new simple proof for the bounds of the right-edge of the support of the free multiplicative convolution µ k , given by Kargin in [5], which show that the support grows at most linearly with k. Moreover, this combinatorial approach generalizes the results of Kargin since we do not require the convolved measures to be identical. We also give further applications, such as a new proof of the limit theorem of Sakuma and Yoshida [11]. * Supported by Deutsche Forschungsgemeinschaft (DFG), Project SP419/8-1.

Singular Values Distribution of Squares of Elliptic Random Matrices and Type B Narayana Polynomials

“…While singular value distributions of powers and products of random matrices with independent entries have been studied extensively (see [8,3,28,22,1,16,10,11,30], just to name a few), the recent results for elliptic random matrices ( [27,26,14]) concern mostly eigenvalue statistics. The obtained asymptotic distribution F is a new generalization of Fuss-Catalan distribution (see [20,5,21,17]). We find moments of this distribution, and prove that its free cumulants are Narayana Polynomials of type B.…”

Simply Generated Non-Crossing Partitions

“…For instance, the number of non-crossing partitions of [n] with given block sizes [28] or the total number of blocks [19] have been studied. Edelman [19] also introduced and enumerated k-divisible non-crossing partitions (where all blocks must have size divisible by k), which have also been studied by Arizmendi and Vargas [4] in connection with free probability. Arizmendi and Vargas also studied k-equal non-crossing partitions (where all blocks must have size exactly k).…”

“…However, the structure of a n may be impacted drastically. To the best of our knowledge, only uniform non-crossing partitions have yet been studied; see [4,12,34].…”

Statistics of Blocks in k-Divisible Non-Crossing Partitions

“…As can be seen in [1] and [2], k-divisible non-crossing partitions play an important role in the calculation of the free cumulants and moments of products of k free random variables. Moreover, in the approach given in [2] for studying asymptotic behavior of the size of the support, when k → ∞, understanding the asymptotic behavior of the sizes of blocks was a crucial step.…”

“…Let us finally mention that the bijections f i are closely related to the Kreweras complement of a (k + 1)-equal non-crossing partitions, which was considered in [2]. Indeed Kr(π) can be divided in a canonical way into k + 1 partitions of [n], π 1 , ..., π k+1 , such that |π i | = |f i (π)|.…”