Asymptotic Behavior of Multiplicative Spherical Integrals and S-transform

TL;DR

This paper analyzes the asymptotic behavior of a multiplicative spherical integral related to the Harish-Chandra Itzykson Zuber integral, revealing connections to the S-transform and eigenvalue distributions, with rigorous proofs and generalizations.

Contribution

It provides a rigorous proof of the asymptotics of a multiplicative spherical integral and generalizes previous results to multiple arguments for eta=1,2.

Findings

Asymptotics involve a modified S-transform of the limit measure.

Rigorous proof of a known result for eta=1,2.

Extension to multiple arguments.

Abstract

In this note, we study the asymptotics of a spherical integral that is a multiplicative counterpart to the well-known Harish-Chandra Itzykson Zuber integral. This counterpart can also be expressed in terms the Heckman-Opdam hypergeometric function. When the argument of this spherical integral is of finite support and of order $N$, these asymptotics involve a modified version of the $S$-transform of the limit measure of the matrix argument and its largest eigenvalue. To prove our main result, we are leveraging a technique of successive conditionning. In particular we prove in a "mathematically rigorous" manner a result from Mergny and Potters in the case $\beta =1,2$ and we generalize it for multiple arguments