A phase transition in the random transposition random walk [article]

Nathanael Berestycki, Rick Durrett (Cornell University)
2004 arXiv   pre-print
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/2∼
more » ... n, where u is an explicit function satisfying u(c)=c/2 for c < 1 and u(c)1. In other words, the distance to the identity is roughly linear during the subcritical phase, and after critical time n/2 it becomes sublinear. In addition, we describe the fluctuations of D_cn/2 about its mean in each of the threeregimes (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 ős-Renyi random graph model.
arXiv:math/0403259v2 fatcat:4xhw6ybribeihmmmlr4iqb4hsq