Asymptotic Behavior of Multiplicative Spherical Integrals and S-transform

2021/08/24 by Husson, Jonathan · 1 citation
#60B20 #60F10 #FOS: Mathematics #Probability (math.PR) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2108.11842

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 β=1,2 and we generalize it for multiple arguments

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• Matrix harmonic analysis at high temperature via the Dirichlet process

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