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

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

2004-12-22

arxiv.org/abs/math/0412449v2

Annals of Applied Probability 2006, Vol. 16, No. 3, 1506-1527

Mathematics

Probability

Published at http://dx.doi.org/10.1214/105051606000000295 in the Annals of Applied Probability (http://www.imstat.org/aap/) by

Scientific paper

10.1214/105051606000000295

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 $\tau (n,L)$, defined as the inverse of the spectral gap of the associated Markov generator, satisfies $\tau (n,L)=O(n\vee \frac{n^3}{L^3})$. 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.

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