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Mixing of Permutations by Biased Transposition

Markov chains defined on the set of permutations of n elements have been studied widely by mathematicians and theoretical computer scientists. We consider chains in which a position i<n is chosen uniformly at random, and then sigma(i) and sigma(i+1) are swapped with probability depending on sigma(i) and sigma(i+1). Our objective is to identify some conditions that assure rapid mixing.

One case of particular interest is what we call the "gladiator chain," in which each number g is assigned a "strength" s_g and when g and g' are swapped, g comes out on top with probability s_g / ( s_g + s_g' ). The stationary probability of this chain is the same as that of the slow-mixing "move ahead one" chain for self-organizing lists, but an open conjecture of Jim Fill's implies that all gladiator chains mix rapidly. Here we obtain some positive partial results by considering cases where the gladiators fall into only a few strength classes.