Spherical functions approach to sums of Random Hermitian Matrices

Abstract

We present an approach to sums of random Hermitian matrices via the theory of spher- ical functions for the Gelfand pair (U(n) Herm(n), U(n)). It is inspired by a similar approach of Kieburg and Kösters for products of random matrices. The spherical func- tions have determinantal expressions because of the Harish-Chandra/Itzykson?Zuber integral formula. It leads to remarkably simple expressions for the spherical transform and its inverse. The spherical transform is applied to sums of unitarily invariant ran- dom matrices from polynomial ensembles and the subclass of polynomial ensembles of derivative type (in the additive sense), which turns out to be closed under addition. We finally present additional detailed calculations for the sum with a random matrix from a Laguerre unitary ensemble.