Regularity Properties of Free Multiplicative Convolution on the Positive Line

Given two nondegenerate Borel probability measures $\mu$ and $\nu$ on ${\mathbb{R}}_{+}=[0,\infty )$, we prove that their free multiplicative convolution $\mu \boxtimes \nu$ has zero singular continuous part and its absolutely continuous part has a density bounded by $x^{-1}$. When $\mu$ and $\nu$ are compactly supported Jacobi measures on $(0,\infty )$ having power law behavior with exponents in $(-1,1)$, we prove that $\mu \boxtimes \nu$ is another Jacobi measure whose density has square root decay at the edges of its support.

Multivariate Fuss-Narayana Polynomials and their Application to Random Matrices

It has been shown recently that the limit moments of $W(n)=B(n)B^{*}(n)$, where $B(n)$ is a product of $p$ independent rectangular random matrices, are certain homogeneous polynomials $P_{k}(d_0,d_1, \ldots , d_{p})$ in the asymptotic dimensions of these matrices. Using the combinatorics of noncrossing partitions, we explicitly determine these polynomials and show that they are closely related to polynomials which can be viewed as {\it multivariate Fuss-Narayana polynomials}. Using this result, we compute the moments of $\varrho_{t_1}\boxtimes \varrho_{t_2}\boxtimes\ldots \boxtimes \varrho_{t_m}$ for any positive $t_1,t_2, \ldots , t_m$, where $\boxtimes$ is the free multiplicative convolution in free probability and $\varrho_{t}$ is the Marchenko-Pastur distribution with shape parameter $t$.