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Probability Theory and Related Fields

, Volume 136, Issue 2, pp 203–233 | Cite as

A phase transition in the random transposition random walk

  • Nathanaël Berestycki
  • Rick Durrett
Article

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 /2u(c)n, where u is an explicit function satisfying u(c)=c/2 for c≤1 and u(c)<c/2 for 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 Erdős-Renyi random graph model.

Keywords or phrases

Random transposition Random graphs Phase transition Coagulation-fragmentation Genome rearrangement Parsimony method 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Nathanaël Berestycki
    • 1
    • 2
  • Rick Durrett
    • 2
  1. 1.Département de Mathématiques et ApplicationsEcole Normale SupérieureParisFrance
  2. 2.Department of MathematicsMalott Hall, Cornell UniversityIthacaUSA

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