Rectangular R-Transform as the Limit of Rectangular Spherical Integrals

Abstract

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 Maïda 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 \(\frac{1}{n}\log\mathbb{E}\{\exp[\sqrt{nm}\theta X_{n}]\}\), where θ∈ℝ, X n is the real part of an entry of U n M n V m and M n is a certain n×m deterministic matrix and U n and V m are independent Haar-distributed orthogonal or unitary matrices with respective sizes n×n and m×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.