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On a surprising relation between the Marchenko-Pastur law, rectangular and square free convolutions

2008/08/28 by Florent Benaych-Georges, Benaych-Georges, Florent
Mathematics · #15A52 #46L54 #Advanced Algebra and Geometry #FOS: Mathematics #Probability (math.PR) #Random Matrices and Applications #Spectral Theory in Mathematical Physics #math.PR #msc:15A52 #msc:46L54

paper · pdf · doi:10.48550/arxiv.0808.3938

11 pages, 1 figure. To appear in Ann. Inst. Henri Poincaré Probab. Stat

openalex publication_date 2008/08/28 · arxiv created 2009/07/09 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

n this paper, we prove a result linking the square and the rectangular R-transforms, the consequence of which is a surprising relation between the square and rectangular versions the free additive convolutions, involving the Marchenko-Pastur law. Consequences on random matrices, on infinite divisibility and on the arithmetics of the square versions of the free additive and multiplicative convolutions are given.

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