Relaxation time of $L$-reversal chains and other chromosome shuffles

Cancrini, N.; Caputo, P.; Martinelli, F.

Relaxation time of $L$-reversal chains and other chromosome shuffles

Authors

Cancrini, N.

Caputo, P.

Martinelli, F.

Description

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 evolution of chromosome chains. The process is described as follows. We have $n$ distinct letters on the vertices of the ${n}$-cycle (${\mathbb{Z}}$ mod $n$); at each step, a connected 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 $τ(n,L)$, defined as the inverse of the spectral gap of the associated Markov generator, satisfies $τ(n,L)=O(n\vee \frac{n^3}{L^3})$. Our results can be interpreted as strong evidence for a conjecture of R. Durrett predicting a similar behavior for the mixing time of the chain.
Published at http://dx.doi.org/10.1214/105051606000000295 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)