Asymptotic Freeness of Random Permutation Matrices with Restricted Cycle Lengths
Abstract
Description
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.
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
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