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February 2021 The interchange process on high-dimensional products
Jonathan Hermon, Justin Salez
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Ann. Appl. Probab. 31(1): 84-98 (February 2021). DOI: 10.1214/20-AAP1583

Abstract

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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Jonathan Hermon. Justin Salez. "The interchange process on high-dimensional products." Ann. Appl. Probab. 31 (1) 84 - 98, February 2021. https://doi.org/10.1214/20-AAP1583

Information

Received: 1 May 2019; Revised: 1 November 2019; Published: February 2021
First available in Project Euclid: 8 March 2021

Digital Object Identifier: 10.1214/20-AAP1583

Subjects:
Primary: 60J10, 60J15, 60J27, 60K35
Secondary: 60B15

Rights: Copyright © 2021 Institute of Mathematical Statistics

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Vol.31 • No. 1 • February 2021
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