Rectangular R-transform at 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 symmetric real matrices, of a result that Guionnet and Mäıda proved for symmetric matrices in [GM05]. More specifically, we study the limit, as n,m tend to infinity, of 1 n logE{exp[nmθXn]}, where Xn is an entry of UnMnVm, θ ∈ R, Mn is a certain n×m deterministic matrix and Un, Vm are independent uniform random orthogonal matrices with respective sizes n × n, m × m. We prove that when the operator norm of Mn is bounded and the singular law of Mn 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 logarithms of Laplace transforms.