Discrete Mathematics & Theoretical Computer Science 
Our work is motivated by BourquePevzner's simulation study of the effectiveness of the parsimony method in studying genome rearrangement, and leads to a surprising result about the random transposition walk in continuous time on the group of permutations on $n$ elements starting from the identity. Let $D_t$ be the minimum number of transpositions needed to go back to the identity element from the location at time $t$. $D_t$ undergoes a phase transition: for $0 < c ≤ 1$, the distance $D_cn/2 ~ cn/2$, i.e., the distance increases linearly with time; for $c > 1$, $D_cn/2 ~ u(c)n$ where u is an explicit function satisfying $u(x) < x/2$. Moreover we describe the fluctuations of $D_{cn/2}$ about its mean at each of the three stages (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 ErdősRényi random graph model.
Source : ScholeXplorer
IsRelatedTo DOI 10.1090/s00029947199411389505 Source : ScholeXplorer IsRelatedTo DOI 10.2307/2154580
