Asymptotic Freeness of Random Permutation Matrices with Restricted Cycle Lengths

• Main Author: Neagu, Mihail G
• Format: Journal Article
• Language: English
• Published: 02-12-2005
• Subjects: Mathematics - Operator Algebras
• DOI: 10.48550/arxiv.math/0512067

Let A_1,A_2,...,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 U_1,U_2,...,U_s of random NxN
permutation matrices with cycle lengths restricted to A_1,A_2,...,A_s,
respectively, converges in *-distribution as N goes to infinity to to a *-free
family u_1,u_2,...,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
A_1,A_2,...,A_s, it is shown that the convergence in *-distribution actually
holds almost surely.