Asymptotic Freeness of Random Permutation Matrices with Restricted Cycle Lengths
Abstract
Let A1,A2,...,As 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,...,Us of random NxN permutation matrices with cycle lengths restricted to A1,A2,...,As, respectively, converges in *-distribution as N goes to infinity to to a *-free family u1,u2,...,us of non-commutative random variables with each ur a Haar unitary (if Ar is an infinite set) or a dr-Haar unitary (if Ar is a finite set and dr=sup Ar).Under an additional assumption on the sets A1,A2,...,As, it is shown that the convergence in *-distribution actually holds almost surely.
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