Rate of Escape of the Mixer Chain

Abstract

The mixer chain on a graph $G$ is the following Markov chain. Place tiles on the vertices of $G$, each tile labeled by its corresponding vertex. A "mixer" moves randomly on the graph, at each step either moving to a randomly chosen neighbor, or swapping the tile at its current position with some randomly chosen adjacent tile. We study the mixer chain on $\mathbb{Z}$, and show that at time $t$ the expected distance to the origin is $t^{3/4}$, up to constants. This is a new example of a random walk on a group with rate of escape strictly between $t^{1/2}$ and $t$.