Asymptotic Behavior of Multiplicative Spherical Integrals and S-transform

Abstract

In this note, we study the asymptotics of an integral on the unitary group first proposed by Mergny and Potters in [10] as a multiplicative counterpart to the well-known Harish-Chandra Itzykson Zuber integral. In particular we prove in a "mathematically rigorous" manner a result from [10] in the case β = 1, 2 and we generalize it for multiple arguments.

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

“…• 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.…”

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

“…Based on ideas develop in [35], we give the precise asymptotic behavior for the annealed free energies of any invariant random matrix, going beyond the case of GOE and Wishart matrices which can tackled by direct Gaussian integration. Combining this result with the asymptotic behavior of the quenched free energy of those three (additive, multiplicative, rectangular) spherical integrals derived in [36][37][38][39][40] allows us to get the rate function in each case.…”