Rectangular R-transform at the limit of rectangular spherical integrals
Résumé
In this paper, proving the analogue, for rectangular or square non symmetric real matrices, of a result that Guionnet and Maida proved for symmetric matrices in 2005, we connect rectangular free probability theory and spherical integrals. More specifically, we study the limit, as $n,m$ tend to infinity, of $\frac{1}{n}\log E\{\exp[\sqrt{nm}\theta X_n]\}$, where $X_n$ is 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 uniform random orthogonal matrices with respective sizes $n\times n$, $m\times m$. We prove that when the operator norm of $M_n$ is bounded and 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 logarithms of Laplace transforms.
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