Asymptotic freeness of random permutation matrices with restricted cycle lengths
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Bibliographic record
Abstract
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 j∈N\Ar 1 j < ∞.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 noncommutative 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.
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| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
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| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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