A Fourier view on the R-transform and related asymptotics of spherical integrals

We estimate the asymptotics of spherical integrals of real symmetric or Hermitian matrices when the rank of one matrix is much smaller than its dimension. 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.