Relaxation time of L-reversal chains and other chromosome shuffles

We prove tight bounds on the relaxation time of the so-called L -reversal chain, which was introduced by R. Durrett as a stochastic model for the evo lution of chromosome chains. The process is described as follows. We have n distinct letters on the vertices of the n-cycle (Z mod n); at each step, a con nected subset of the graph is chosen uniformly at random among all those of length at most L, and the current permutation is shuffled by reversing the order of the letters over that subset. We show that the relaxation time z(n,L), defined as the inverse of the spectral gap of the associated Markov genera