Asymptotic behavior of the multiplicative counterpart of the Harish-Chandra integral and the $S$-transform

Abstract

In this note, we study the asymptotic of spherical integrals, which are analytical extension in index of the normalized Schur polynomials for $\beta =2$ , and of Jack symmetric polynomials otherwise. Such integrals are the multiplicative counterparts of the Harish-Chandra-Itzykson-Zuber (HCIZ) integrals, whose asymptotic are given by the so-called $R$-transform when one of the matrix is of rank one. We argue by a saddle-point analysis that a similar result holds for all $\beta >0$ in the multiplicative case, where the asymptotic is governed by the logarithm of the $S$-transform. As a consequence of this result one can calculate the asymptotic behavior of complete homogeneous symmetric polynomials.

arXiv comment14 pages, 1 figure

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Cited by in corpus (4)

• Rank one HCIZ at high temperature: interpolating between classical and free convolutions
• Large Deviations Asymptotics of Rectangular Spherical Integral
• Stochastic and Quantum Dynamics of Repulsive Particles: from Random Matrix Theory to Trapped Fermions
• Asymptotic Behavior of Multiplicative Spherical Integrals and S-transform