Asymptotic Freeness of Random Permutation Matrices with Restricted Cycle Lengths

2005/12/02 by Mihail G. Neagu, Neagu, Mihail G.
Mathematics · #05A16 (secondary) #15A52 #46L54 (primary) #FOS: Mathematics #Operator Algebras (math.OA) #math.OA #msc:05A16 #msc:15A52 #msc:46L54

paper · pdf · doi:10.48550/arxiv.math/0512067

27 pages, 2 figures; substantial modifications were made (especially to section 4), leading to a new almost sure convergence result stated as part of the main theorem

arxiv created 2007/06/26 · arxiv updated 2009/12/01

Abstract

Let A₁,A₂,...,A_s be a finite sequence of (not necessarily disjoint, or even distinct) non-empty sets of positive integers satisfying a certain condition. It is shown that an independent family U₁,U₂,...,U_s of random NxN permutation matrices with cycle lengths restricted to A₁,A₂,...,A_s, respectively, converges in *-distribution as N goes to infinity to to a *-free family u₁,u₂,...,u_s of non-commutative random variables with each u_r a Haar unitary (if A_r is an infinite set) or a d_r-Haar unitary (if A_r is a finite set and d_r=sup A_r).Under an additional assumption on the sets A₁,A₂,...,A_s, it is shown that the convergence in *-distribution actually holds almost surely.

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