Portal de Periódicos da CAPES

The interchange process on high-dimensional products

2021; Institute of Mathematical Statistics; Volume: 31; Issue: 1 Linguagem: Inglês

10.1214/20-aap1583

2168-8737

Jonathan Hermon, Justin Salez,

Limits and Structures in Graph Theory

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).