Spherical functions approach to sums of random Hermitian matrices

TL;DR

This paper introduces a spherical functions approach to analyze sums of random Hermitian matrices, providing explicit formulas and applications to polynomial ensembles and Laguerre Unitary Ensembles.

Contribution

It develops a novel spherical functions framework for sums of random Hermitian matrices, extending previous methods and enabling explicit calculations for polynomial ensembles.

Findings

Derived simple expressions for spherical transforms and their inverses.

Showed polynomial ensembles of derivative type are closed under addition.

Applied the method to sums involving Laguerre Unitary Ensemble matrices.

Abstract

We present an approach to sums of random Hermitian matrices via the theory of spherical functions for the Gelfand pair $(\mathrm{U}(n) \ltimes \mathrm{Herm}(n), \mathrm{U}(n))$. It is inspired by a similar approach of Kieburg and K\"osters for products of random matrices. The spherical functions 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 random 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.