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 $t$.
@article{10_37236_2799,
author = {Romuald Lenczewski and Rafal Salapata},
title = {Multivariate {Fuss-Narayana} polynomials and their application to random matrices},
journal = {The electronic journal of combinatorics},
year = {2013},
volume = {20},
number = {2},
doi = {10.37236/2799},
zbl = {1267.05024},
url = {http://geodesic.mathdoc.fr/articles/10.37236/2799/}
}
TY - JOUR
AU - Romuald Lenczewski
AU - Rafal Salapata
TI - Multivariate Fuss-Narayana polynomials and their application to random matrices
JO - The electronic journal of combinatorics
PY - 2013
VL - 20
IS - 2
UR - http://geodesic.mathdoc.fr/articles/10.37236/2799/
DO - 10.37236/2799
ID - 10_37236_2799
ER -
%0 Journal Article
%A Romuald Lenczewski
%A Rafal Salapata
%T Multivariate Fuss-Narayana polynomials and their application to random matrices
%J The electronic journal of combinatorics
%D 2013
%V 20
%N 2
%U http://geodesic.mathdoc.fr/articles/10.37236/2799/
%R 10.37236/2799
%F 10_37236_2799
Romuald Lenczewski; Rafal Salapata. Multivariate Fuss-Narayana polynomials and their application to random matrices. The electronic journal of combinatorics, Tome 20 (2013) no. 2. doi: 10.37236/2799
