Existence of a phase transition of the interchange process on the Hamming graph

Abstract

The interchange process on a finite graph is obtained by placing a particle on each vertex of the graph, then at rate $1$, selecting an edge uniformly at random and swapping the two particles at either end of this edge. In this paper we develop new techniques to show the existence of a phase transition of the interchange process on the $2$-dimensional Hamming graph. We show that in the subcritical phase, all of the cycles of the process have length $O(\log n)$, whereas in the supercritical phase a positive density of vertices lies in cycles of length at least $n^{2-\varepsilon }$ for any $\varepsilon >0$.