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.


  1. 1.
    Aizenman, M.: Geometric analysis of φ 4 fields and Ising models. Commun. Math. Phys. 86, 1–48 (1982) MathSciNetADSMATHCrossRefGoogle Scholar
  2. 2.
    Aizenman, M., Nachtergaele, B.: Geometric aspects of quantum spin states. Commun. Math. Phys. 164, 17–63 (1994) MathSciNetADSMATHCrossRefGoogle Scholar
  3. 3.
    Aldous, D.: Deterministic and stochastic models for coalescence (aggregation and coagulation): a review of the mean-field theory for probabilists. Bernoulli 5, 3–48 (1999) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Arratia, R., Barbour, A.D., Tavaré, S.: Logarithmic Combinatorial Structures: A Probabilistic Approach. EMS Monographs in Mathematics. Eur. Math. Soc., Zürich (2003) CrossRefGoogle Scholar
  5. 5.
    Berestycki, N.: Emergence of giant cycles and slowdown transition in random transpositions and k-cycles. Electron. J. Probab. 16, 152–173 (2011) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Berestycki, N., Durrett, R.: Limiting behavior for the distance of a random walk. Electron. J. Probab. 13, 374–395 (2008) MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Bertoin, J.: Random Fragmentation and Coagulation Processes. Cambridge University Press, Cambridge (2006) MATHCrossRefGoogle Scholar
  8. 8.
    Betz, V., Ueltschi, D.: Spatial random permutations and infinite cycles. Commun. Math. Phys. 285, 469–501 (2009) MathSciNetADSMATHCrossRefGoogle Scholar
  9. 9.
    Betz, V., Ueltschi, D.: Spatial random permutations and Poisson-Dirichlet law of cycle lengths. Electron. J. Probab. 16, 1173–1192 (2011) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Crawford, N., Ioffe, D.: Random current representation for transverse field Ising model. Commun. Math. Phys. 296, 447–474 (2010) MathSciNetADSMATHCrossRefGoogle Scholar
  11. 11.
    Diaconis, P., Mayer-Wolf, E., Zeitouni, O., Zerner, M.P.W.: The Poisson-Dirichlet law is the unique invariant distribution for uniform split-merge transformations. Ann. Probab. 32, 915–938 (2004) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Feng, S.: The Poisson-Dirichlet Distribution and Related Topics. Probability and Its Applications, Springer, Berlin (2010) MATHCrossRefGoogle Scholar
  13. 13.
    Feynman, R.P.: Atomic theory of the λ transition in Helium. Phys. Rev. 91, 1291–1301 (1953) MathSciNetADSMATHCrossRefGoogle Scholar
  14. 14.
    Grimmett, G.: Space-time percolation. In: In and Out of Equilibrium. 2. Prog. Probab., vol. 60, pp. 305–320. Birkhäuser, Basel (2008) CrossRefGoogle Scholar
  15. 15.
    Gandolfo, D., Ruiz, J., Ueltschi, D.: On a model of random cycles. J. Stat. Phys. 129, 663–676 (2007) MathSciNetADSMATHCrossRefGoogle Scholar
  16. 16.
    Goldschmidt, C., Ueltschi, D., Windridge, P.: Quantum Heisenberg models and their probabilistic representations. In: Entropy and the Quantum II. Contemporary Mathematics, vol. 552, pp. 177–224. Am. Math. Soc., Providence (2011). arXiv:1104.0983 CrossRefGoogle Scholar
  17. 17.
    Kerl, J.: Shift in critical temperature for random spatial permutations with cycle weights. J. Stat. Phys. 140, 56–75 (2010) MathSciNetADSMATHCrossRefGoogle Scholar
  18. 18.
    Kingman, J.F.C.: Random discrete distributions. J. R. Stat. Soc. B 37, 1–15 (1975) MathSciNetMATHGoogle Scholar
  19. 19.
    Kingman, J.F.C.: Mathematics of Genetic Diversity. CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 34. SIAM, Philadelphia (1980) CrossRefGoogle Scholar
  20. 20.
    Pitman, J., Yor, M.: The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator. Ann. Probab. 25, 855–900 (1997) MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Pitman, J.: Poisson-Dirichlet and GEM invariant distributions for split-and-merge transformations of an interval partition. Comb. Probab. Comput. 11, 501–514 (2002) MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Schramm, O.: Compositions of random transpositions. Isr. J. Math. 147, 221–243 (2005) MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Shepp, L.A., Lloyd, S.L.: Ordered cycle lengths in a random permutation. Trans. Am. Math. Soc. 121, 340–357 (1966) MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Sütő, A.: Percolation transition in the Bose gas. J. Phys. A 26, 4689–4710 (1993) MathSciNetADSCrossRefGoogle Scholar
  25. 25.
    Tóth, B.: Improved lower bound on the thermodynamic pressure of the spin 1/2 Heisenberg ferromagnet. Lett. Math. Phys. 28, 75 (1993) MathSciNetADSMATHCrossRefGoogle Scholar
  26. 26.
    Tsilevich, N.V.: Stationary random partitions of a natural series. Teor. Veroyatnost. i Primenen. 44, 55–73 (1999) MathSciNetGoogle Scholar
  27. 27.
    Wilson, D.: Mixing times of lozenge tiling and card shuffling Markov chains. Ann. Appl. Probab. 14, 274–325 (2004) MathSciNetMATHCrossRefGoogle Scholar

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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