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Probability Theory and Related Fields

, Volume 139, Issue 1–2, pp 143–189 | Cite as

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

  • Florent Benaych-Georges
Article

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.

Keywords

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

Mathematics Subject Classifications (2000)

15A52 46L54 60E07 60F05 

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Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.DMA, École Normale SupérieureParis Cedex 05France

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