A phase transition in the random transposition random walk

Nathanael Berestycki (1) , Rick Durrett (2)

1 Department of Mathematics [Cornell]
2 ENS-PSL - École normale supérieure - Paris

Our work is motivated by Bourque-Pevzner'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 coagulation-fragmentation process and relating the behavior to the Erdős-Rényi random graph model.

Mots clés: random transposition; random graphs; phase transition; coagulation-fragmentation

Domaines: Algorithme et structure de données [cs.DS]; Mathématique discrète [cs.DM]; Combinatoire [math.CO]; Géométrie algorithmique [cs.CG]

Nathanael Berestycki, Rick Durrett. A phase transition in the random transposition random walk. Discrete Random Walks, DRW'03, 2003, Paris, France. pp.17-26, ⟨10.46298/dmtcs.3343⟩. ⟨hal-00001309v3⟩