Rectangular R-transform as the limit of rectangular spherical integrals

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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 $\mu$.