article

Title: Asymptotic freeness of random permutation matrices with restricted cycle lengths

Authors: Mihail G. Neagu

Issue: Volume 56 (2007), Issue 4, 2017-2049

Abstract:

$Let $A_1, A_2, \dots, A_s$ 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 $\sum_{j \in \mathbb{N} \setminus A_r} 1/j < \infty$. It is shown that an independent family $U_1, U_2, \dots, U_s$ of uniformly distributed random $N \times N$ permutation matrices with cycle lengths restricted to $A_1, A_2, \dots, A_s$, respectively, converges in $*$-distribution as $N \to \infty$ to a $*$-free family $u_1, u_2, \dots, u_s$ 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 $A_1, A_2, \dots, A_s$ either consists of a single positive integer or is infinite, it is shown that the convergence in $*$-distribution actually holds almost surely.$