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

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dc.creatorCancrini, N.
dc.creatorCaputo, P.
dc.creatorMartinelli, F.
dc.date2004-12-22
dc.date2006-10-11
dc.date.accessioned2026-07-07T06:39:12Z
dc.date.available2026-07-07T06:39:12Z
dc.descriptionWe 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.
dc.descriptionPublished 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)
dc.identifierhttps://arxiv.org/abs/math/0412449
dc.identifierhttp://arxiv.org/abs/math/0412449
dc.identifierAnnals of Applied Probability 2006, Vol. 16, No. 3, 1506-1527
dc.identifierdoi:10.1214/105051606000000295
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/100991
dc.subjectProbability
dc.subject60J27, 92D10 (Primary)
dc.titleRelaxation time of $L$-reversal chains and other chromosome shuffles
dc.typetext

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