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

Lacoin, Hubert ^[1]
1. [1] Instituto Nacional de Matemática Pura e Aplicada
  
  Instituto Nacional de Matemática Pura e Aplicada
  
  Brasil
Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 44, Nº. 2, 2016, págs. 1426-1487
Idioma: inglés
DOI: 10.1214/15-AOP1004
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Resumen
- In this paper, we investigate the mixing time of the adjacent transposition shuffle for a deck of N cards. We prove that around time N2logN/(2π2), 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 (N2logN)/π2. 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.