2009/09/01 by Florent Benaych-Georges, Benaych-Georges, Florent · 2 citations
Mathematics · #FOS: Mathematics #Mathematical Inequalities and Applications #Mathematical functions and polynomials #Probability (math.PR) #Random Matrices and Applications
paper · doi:10.48550/arxiv.0909.0178
openalex publication_date 2009/09/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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[√(nm)θXn], where Xn is the real part of an entry of Un Mn Vm, θ is a real number, 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 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 log-Laplace transforms.