Rectangular R-Transform as the Limit of Rectangular Spherical Integrals

Florent Benaych-Georges ()
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Florent Benaych-Georges: UPMC Univ. Paris 6

Journal of Theoretical Probability, 2011, vol. 24, issue 4, 969-987

Abstract: 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.

Keywords: Random matrices; Free probability; Rectangular R-transform; Haar measure; Spherical integrals; 15A52; 46L54; 60B15 (search for similar items in EconPapers)
Date: 2011
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DOI: 10.1007/s10959-011-0362-7

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