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 β = 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 β > 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.

Right large deviation principle for the top eigenvalue of the sum or product of invariant random matrices

“…Note that if we do the change of variable {s i (A)} → {λ i AA T } given by Eq. (38) in the joint law of Eq. ( 42), we have that the matrix AA T is taken from an invariant ensemble with the modified potential Ṽq (.)…”

“…• On the one hand, the quenched free energy associated to each spherical integral has been computed before in the literature, see Refs. [36,37,38,39,40], and is known to satisfy a transition depending on the parameter θ, between a phase where it does not depend explicitly on the position of the top eigenvalue/singular value and a phase where it does. The asymptotics of the quenched free energy is summarized in Sec.…”

“…The behavior of Z C (θ) for N large has been recently investigated in Refs. [38,39] and are given in the following section. Due to its similarity with the original SSK model, one should expect to have a similar behavior, with a paramagnetic phase at high temperature and a spin glass phase at low temperature.…”

“…In the high temperature regime, it has been shown in Refs. [38,39] that the derivative with respect to the parameter θ of the quenched free energy is for small enough θ the logarithm of the S-transform. One can also get the behavior for low temperature where there is a saturation, and we have:…”

“…Combining this result with the asymptotic behavior of the quenched free energy of those three (additive, multiplicative, rectangular) spherical integrals derived in Refs. [36,37,38,39,40] allows us to get the rate function in each case.…”