Rate of Escape of the Mixer Chain

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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 t³/⁴, up to constants. This is a new example of a random walk on a group with rate of escape strictly between t¹/² and t.