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Mixing time and cutoff for the adjacent transposition shuffle and the simple exclusion

Lacoin, Hubert (2016), Mixing time and cutoff for the adjacent transposition shuffle and the simple exclusion, Annals of Probability, 44, 2, p. 1426-1487. 10.1214/15-AOP1004

Type
Article accepté pour publication ou publié
Date
2016
Journal name
Annals of Probability
Volume
44
Number
2
Publisher
Institute of Mathematical Statistics
Published in
Paris
Pages
1426-1487
Publication identifier
Metadata
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Author(s)
Lacoin, Hubert
Abstract (EN)
In this paper, we investigate the mixing time of the adjacent transposition shuffle for a deck of cards. We prove that around time N^2\log N/(2\pi^2), the total-variation distance to equilibrium of the deck distribution drops abruptly from 1 to 0, and that the separation distance has a similar behavior but with a transition occurring at time (N^2\log N)/\pi^2. This solves a conjecture formulated by David Wilson. We present also similar results for the exclusion process on a segment of length N with k particles.
Subjects / Keywords
Shuffle; Mixing time; Particle systems; Cutoff; Markov chains