An asymptotic log-Fourier interpretation of the R-transform

Details An asymptotic log-Fourier interpretation of the R-transform An asymptotic log-Fourier interpretation of the R-transform

2004-06-07

arxiv.org/abs/math/0406121v3

Mathematics

Probability

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

Scientific paper

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.

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