Rate of escape of the mixer chain

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 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.