The interchange process on high-dimensional products

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).