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Media type:
E-Article
Title:
Asymptotic Freeness of Random Permutation Matrices with Restricted Cycle Lengths
Contributor:
Neagu, Mihail G.
Published:
Department of Mathematics of Indiana University, 2007
Published in:
Indiana University Mathematics Journal, 56 (2007) 4, Seite 2017-2049
Language:
English
ISSN:
0022-2518;
1943-5258
Origination:
Footnote:
Description:
Let A1, A2,..., As be a finite sequence of (not necessarily disjoint, or even distinct) non-empty sets of positive integers such that each Ar either is a finite set or satisfies $\sum _{\mathrm{j}\in \mathrm{\mathbb{N}}\backslash {\mathrm{A}}_{\mathrm{r}}}1/\mathrm{j}<\mathrm{\infty }$. It is shown that an independent family U1, U2,..., Us of uniformly distributed random N × N permutation matrices with cycle lengths restricted to A1, A2,..., As, respectively, converges in ∗-distribution as N → ∞ to a ∗-free family u1, u2,..., us of non-commutative random variables, where each ur is a (max Ar)-Haar unitary (if Ar is a finite set) or a Haar unitary (if Ar is an infinite set). Under the additional assumption that each of the sets A1, A2,..., As either consists of a single positive integer or is infinite, it is shown that the convergence in ∗-distribution actually holds almost surely.