Publications [#243548] of Richard T. Durrett

Papers Published

1. Berestycki, N; Durrett, R, A phase transition in the random transposition random walk, Probability Theory and Related Fields, vol. 136 no. 2 (January, 2006), pp. 203-233, Springer Nature, ISSN 0178-8051
   (last updated on 2024/04/18)

   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)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 coagulation-fragmentation process and relating the behavior to the Erdos-Renyi random graph model.