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The interchange process on high-dimensional products

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1905.02146.pdf (229.6Kb)
Date
2021-02
Dewey
Probabilités et mathématiques appliquées
Sujet
comparison of Dirichlet forms; interchange process; Mixing times; product graphs
Journal issue
Annals of Applied Probability
Volume
31
Number
1
Publication date
02-2021
Article pages
84-98
Publisher
Institute of Mathematical Statistics
DOI
http://dx.doi.org/10.1214/20-AAP1583
URI
https://basepub.dauphine.fr/handle/123456789/21654
Collections
  • CEREMADE : Publications
Metadata
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Author
Hermon, Jonathan
128731 Department of mathematics, University of British Columbia
Salez, Justin
60 CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Type
Article accepté pour publication ou publié
Abstract (EN)
We resolve a long-standing conjecture of Wilson (Ann. Appl. Probab.14 (2004) 274–325), reiterated by Oliveira (2016), asserting that the mixing time of the interchange process with unit edge rates on the n-dimensional hypercube is of order n. This follows from a sharp inequality established at the level of Dirichlet forms, from which we also deduce that macroscopic cycles emerge in constant time, and that the log-Sobolev constant of the exclusion process is of order 1. Beyond the hypercube, our results apply to cartesian products of arbitrary graphs of fixed size, shedding light on a broad conjecture of Oliveira (Ann. Probab.41 (2013) 871–913).

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