Rectangular R-transform as the limit of rectangular spherical integrals

In this paper, we connect rectangular free probability theory and spherical integrals. We prove the analogue, for rectangular or square non-Hermitian matrices, of a result that Guionnet and Mäıda proved for Hermitian matrices in [12]. More specifically, we study the limit, as n,m tend to infinity, of 1 n logE{exp[nmθXn]}, where θ ∈ R, Xn is the real part of an entry of UnMnVm, Mn is a certain n×m deterministic matrix and Un, Vm are independent Haar-distributed orthogonal or unitary matrices with respective sizes n×n, m×m. We prove that when the singular law ofMn 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.