# The interchange process on high-dimensional products

## TL;DR

This paper proves that the mixing time of the interchange process on high-dimensional hypercubes is proportional to the dimension, showing rapid emergence of macroscopic cycles and providing bounds on related constants.

## Contribution

It resolves a long-standing conjecture about the mixing time on hypercubes and extends results to products of arbitrary fixed-size graphs.

## Key findings

- Mixing time on hypercube is of order n
- Macroscopic cycles emerge in constant time
- Log-Sobolev constant is of order 1

## Abstract

We resolve a long-standing conjecture of Wilson (2004), reiterated by Oliveira (2016), asserting that the mixing-time of the unit-rate Interchange Process 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 (2013).