Asymptotic freeness of random permutation matrices with restricted cycle lengths

Abstract

Let A 1, A 2 , ... , As be a finite sequence of (not necessarily disjoint, or even distinct) non-empty sets of positive integers such that each A r either is a finite set or satisfies Σ j∈N\Ar 1/j < ∞, It is shown that an independent family U 1 U 2 ,..., U s of uniformly distributed random N x N permutation matrices with cycle lengths restricted to A 1 , A 2 ,..., A s , respectively, converges in *-distribution as N - oo to a *-free family u 1 , u 2 ,..., u s of non-commutative random variables, where each u r is a (max A r )-Haar unitary (if A r is a finite set) or a Haar unitary (if A r is an infinite set). Under the additional assumption that each of the sets A 1 , A 2 ,..., As either consists of a single positive integer or is infinite, it is shown that the convergence in * -distribution actually holds almost surely.