Journal of Statistical Physics

, Volume 146, Issue 6, pp 1105–1121 | Cite as

Lattice Permutations and Poisson-Dirichlet Distribution of Cycle Lengths

  • Stefan Grosskinsky
  • Alexander A. Lovisolo
  • Daniel Ueltschi


We study random spatial permutations on ℤ3 where each jump xπ(x) is penalized by a factor \(\mathrm{e}^{-T\| x-\pi (x)\|^{2}}\). The system is known to exhibit a phase transition for low enough T where macroscopic cycles appear. We observe that the lengths of such cycles are distributed according to Poisson-Dirichlet. This can be explained heuristically using a stochastic coagulation-fragmentation process for long cycles, which is supported by numerical data.


Lattice permutations Cycle lengths Poisson-Dirichlet distribution 



D.U. is grateful to N. Berestycki, A. Hammond, and J. Martin for useful discussions. A.A.L. was funded by the Erasmus Mundus Masters Course CSSM. S.G. and D.U. acknowledge support by EPSRC, grants no. EP/E501311/1 and EP/G056390/1, respectively.


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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Stefan Grosskinsky
    • 1
    • 2
  • Alexander A. Lovisolo
    • 1
  • Daniel Ueltschi
    • 2
  1. 1.Centre for Complexity ScienceUniversity of WarwickCoventryUK
  2. 2.Department of MathematicsUniversity of WarwickCoventryUK

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