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Relaxation time of L-reversal chains and other chromosome shuffles | | Nicoletta Cancrini
; Pietro Caputo
; Fabio Martinelli
; | | Date: |
22 Dec 2004 | | Subject: | Probability | math.PR | | Abstract: | We prove tight bounds on the relaxation time of the so called $L$--reversal chain, 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 ($bZ$ 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 $ (n,L)$, defined as the inverse of the spectral gap of the associated Markov generator, satisfies $ (n,L)=O(n vee frac{n^3}{L^3})$. Our results can be interpreted as a strong evidence for a conjecture of R. Durrett of a similar behavior for the mixing time of the chain. | | Source: | arXiv, math.PR/0412449 | | Services: | Forum | Review | PDF | Favorites | |
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