Rectangular R-transform as the limit of rectangular spherical integrals

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 $\exp[\sqrt{nm}θX_n]$, where $X_n$ is the real part of an entry of $U_n M_n V_m$, $θ$ 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 $μ$, for $θ$ small enough, this limit actually exists and can be expressed with the rectangular R-transform of $μ$. 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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