Rectangular R-transform as the limit of rectangular spherical integrals

Abstract : In this paper, we connect rectangular free probability theory and spherical integrals. In this way, we prove the analogue, for rectangular or square non-Hermitian matrices, of a result that Guionnet and Maida proved for Hermitian matrices in 2005. More specifically, we study the limit, as $n,m$ tend to infinity, of the logarithm (divided by $n$) of the expectation of $\exp[\sqrt{nm}\theta X_n]$, where $X_n$ is the real part of an entry of $U_n M_n V_m$, $\theta$ is a real number, $M_n$ is a certain $n\times m$ deterministic matrix and $U_n, V_m$ are independent Haar-distributed orthogonal or unitary matrices with respective sizes $n\times n$, $m\times m$. We prove that when the singular law of $M_n$ converges to a probability measure $\mu$, for $\theta$ small enough, this limit actually exists and can be expressed with the rectangular R-transform of $\mu$. This gives an interpretation of this transform, which linearizes the rectangular free convolution, as the limit of a sequence of log-Laplace transforms.
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Preprints, Working Papers, ...
17 pages. 2009
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https://hal.archives-ouvertes.fr/hal-00412364
Contributor : Florent Benaych-Georges <>
Submitted on : Tuesday, April 19, 2011 - 8:38:25 PM
Last modification on : Monday, May 29, 2017 - 2:25:19 PM
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  • HAL Id : hal-00412364, version 7
  • ARXIV : 0909.0178

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Florent Benaych-Georges. Rectangular R-transform as the limit of rectangular spherical integrals. 17 pages. 2009. <hal-00412364v7>

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