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arXiv · 2108.11842

Asymptotic Behavior of Multiplicative Spherical Integrals and S-transform

Jonathan Husson
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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

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Jonathan Husson. 2021-08-24. Asymptotic Behavior of Multiplicative Spherical Integrals and S-transform. https://arxiv.org/abs/2108.11842

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