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Conference Papers Discrete Mathematics and Theoretical Computer Science Year : 2003

A phase transition in the random transposition random walk

Nathanael Berestycki
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Rick Durrett
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Abstract

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.
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Dates and versions

hal-00001309 , version 1 (16-03-2004)
hal-00001309 , version 2 (13-10-2004)
hal-00001309 , version 3 (12-08-2015)

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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⟩
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