Asymptotic Freeness of Random Permutation Matrices with Restricted Cycle Lengths

Details Asymptotic Freeness of Random Permutation Matrices with Restricted Cycle Lengths Asymptotic Freeness of Random Permutation Matrices with Restricted Cycle Lengths

2005-12-02

arxiv.org/abs/math/0512067v2

Mathematics

Operator Algebras

27 pages, 2 figures; substantial modifications were made (especially to section 4), leading to a new almost sure convergence r

Scientific paper

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.

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