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An asymptotic log-Fourier interpretation of the R-transform

Alice Guionnet·Mylene Maida
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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.

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Alice Guionnet, Mylene Maida. 2005-03-11. An asymptotic log-Fourier interpretation of the R-transform. https://arxiv.org/abs/math/0406121

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