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Rectangular R-Transform as the Limit of Rectangular Spherical Integrals

Author

Listed:
  • Florent Benaych-Georges

    (UPMC Univ. Paris 6
    École Polytechnique)

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.

Suggested Citation

  • Florent Benaych-Georges, 2011. "Rectangular R-Transform as the Limit of Rectangular Spherical Integrals," Journal of Theoretical Probability, Springer, vol. 24(4), pages 969-987, December.
    • Handle: RePEc:spr:jotpro:v:24:y:2011:i:4:d:10.1007_s10959-011-0362-7
    DOI: 10.1007/s10959-011-0362-7
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