Rectangular R-transform at the limit of rectangular spherical integrals
Résumé
In this paper, we connect rectangular free probability theory and spherical integrals. In this way, we prove the analogue, for rectangular or square non symmetric real matrices, of a result that Guionnet and Maïda proved for symmetric matrices. 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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