Rectangular R-transform as the limit of rectangular spherical integrals
Résumé
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 Ma < da proved for Hermitian matrices in (J. Funct. Anal. 222(2):435-490, 2005). More specifically, we study the limit, as n and m tend to infinity, of 1/nlog E{exp[root nm theta X-n]}, where theta aa"e, X (n) is the real part of an entry of U (n) M (n) V (m) and M (n) is a certain nxm deterministic matrix and U (n) and V (m) are independent Haar-distributed orthogonal or unitary matrices with respective sizes nxn and mxm. 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. This gives an interpretation of this transform, which linearizes the rectangular free convolution, as the limit of a sequence of log-Laplace transforms.