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Infinitely divisible distributions for rectangular free convolution: classification and matricial interpretation

Abstract

In a previous paper (Benaych-Georges in Related Convolution 2006), we defined the rectangular free convolution ⊞λ. Here, we investigate the related notion of infinite divisibility, which happens to be closely related the classical infinite divisibility: there exists a bijection between the set of classical symmetric infinitely divisible distributions and the set of ⊞λ -infinitely divisible distributions, which preserves limit theorems. We give an interpretation of this correspondence in terms of random matrices: we construct distributions on sets of complex rectangular matrices which give rise to random matrices with singular laws going from the symmetric classical infinitely divisible distributions to their ⊞λ-infinitely divisible correspondents when the dimensions go from one to infinity in a ratio λ.

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Correspondence to Florent Benaych-Georges.

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Benaych-Georges, F. Infinitely divisible distributions for rectangular free convolution: classification and matricial interpretation. Probab. Theory Relat. Fields 139, 143–189 (2007). https://doi.org/10.1007/s00440-006-0042-1

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Keywords

  • Random matrices
  • Free probability
  • Free convolution
  • Marchenko–Pastur distribution
  • Infinitely divisible distributions

Mathematics Subject Classifications (2000)

  • 15A52
  • 46L54
  • 60E07
  • 60F05