Strong asymptotic freeness for Wigner and Wishart matrices

2005/04/20 by Mireille Capitaine, Capitaine, Mireille, Catherine Donati-Martin +1 · 3 citations
Mathematics · #15A52 #46L54 #60F99 #FOS: Mathematics #Probability (math.PR) #math.PR #msc:15A52 #msc:46L54 #msc:60F99

paper · pdf · doi:10.48550/arxiv.math/0504414

arxiv created 2005/04/20 · arxiv updated 2009/12/01

Abstract

We prove that any non commutative polynomial of r independent copies of Wigner matrices converges a.s. towards the polynomial of r free semicircular variables in operator norm. This result extends a previous work of Haagerup and Thorbjornsen where GUE matrices are considered, as well as the classical asymptotic freeness for Wigner matrices (i.e. convergence of the moments) proved by Dykema. We also study the Wishart case.

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