An asymptotic log-Fourier interpretation of the R-transform

Abstract

We estimate the asymptotics of spherical integrals when the rank of one matrix is finite. We show that it is given in terms of the R-transform of the spectral measure of the full rank matrix and give a new proof of the fact that the R-transform is additive under free convolution. These asymptotics also extend to the case where one matrix has rank one but complex eigenvalue, a result related with the analyticity of the corresponding spherical integrals.

Revised version : the proof for the full asymptotics in rank one has been corrected and a bit shortened. An appendix was added