Strong asymptotic freeness for Wigner and Wishart matrices

Mireille Capitaine (1) , Catherine Donati-Martin (2)

1 LSP - Laboratoire de Statistique et Probabilités
2 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

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.

Mots clés: random matrices; free probability; asymptotic freeness

Domaines: Probabilités [math.PR]

Mireille Capitaine, Catherine Donati-Martin. Strong asymptotic freeness for Wigner and Wishart matrices. 2005. ⟨hal-00004770⟩