Math @ Duke

Publications [#243548] of Richard T. Durrett
Papers Published
 Berestycki, N; Durrett, R, A phase transition in the random transposition random walk,
Probability Theory and Related Fields, vol. 136 no. 2
(2006),
pp. 203233, ISSN 01788051 [MR2240787 (2007i:60009)], [doi]
(last updated on 2017/12/11)
Abstract: Our work is motivated by Bourque and Pevzner's (2002) simulation study of the effectiveness of the parsimony method in studying genome rearrangement, and leads to a surprising result about the random transposition walk on the group of permutations on n elements. Consider this walk in continuous time starting at the identity and let D t be the minimum number of transpositions needed to go back to the identity from the location at time t. D t undergoes a phase transition: the distance D cn/2̃ u(c)n, where u is an explicit function satisfying u(c)=c/2 for c ≤ 1 and u(c)<c/2 for c>1. In addition, we describe the fluctuations of D cn/2 about its mean in each of the three regimes (subcritical, critical and supercritical). The techniques used involve viewing the cycles in the random permutation as a coagulationfragmentation process and relating the behavior to the ErdosRenyi random graph model.


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