Electronic Journal of Probability

Spatial Random Permutations and Poisson-Dirichlet Law of Cycle Lengths

Volker Betz and Daniel Ueltschi

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We study spatial permutations with cycle weights that are bounded or slowly diverging. We show that a phase transition occurs at an explicit critical density. The long cycles are macroscopic and their cycle lengths satisfy a Poisson-Dirichlet law.

Article information

Electron. J. Probab., Volume 16 (2011), paper no. 41, 1173-1192.

Accepted: 6 June 2011
First available in Project Euclid: 1 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82B26: Phase transitions (general)

Spatial random permutations cycle weights Poisson-Dirichlet distribution

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Betz, Volker; Ueltschi, Daniel. Spatial Random Permutations and Poisson-Dirichlet Law of Cycle Lengths. Electron. J. Probab. 16 (2011), paper no. 41, 1173--1192. doi:10.1214/EJP.v16-901. https://projecteuclid.org/euclid.ejp/1464820210

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  • Arratia, Richard; Barbour, A. D.; Tavaré, Simon. Random combinatorial structures and prime factorizations. Notices Amer. Math. Soc. 44 (1997), no. 8, 903–910.
  • Berestycki. N, Emergence of giant cycles and slowdown transition in random transpositions and $k$-cycles, Electr. J. Probab. 16, 152–173 (2011)
  • Betz, Volker; Ueltschi, Daniel. Spatial random permutations and infinite cycles. Comm. Math. Phys. 285 (2009), no. 2, 469–501.
  • Betz. V, Ueltschi. D. Spatial random permutations with small cycle weights}, Probab. Th. Rel. Fields 149, 191–222 (2011)
  • Betz. V, Ueltschi. D, Critical temperature of dilute Bose gases, Phys. Rev. A 81, 023611 (2010)
  • Betz, Volker; Ueltschi, Daniel; Velenik, Yvan. Random permutations with cycle weights. Ann. Appl. Probab. 21 (2011), no. 1, 312–331.
  • Biskup. M, Richthammer. T. in preparation
  • Buffet, E.; Pulé, J. V. Fluctuation properties of the imperfect Bose gas. J. Math. Phys. 24 (1983), no. 6, 1608–1616.
  • Ewens, W. J. The sampling theory of selectively neutral alleles. Theoret. Population Biology 3 (1972), 87–112; erratum, ibid. 3 (1972), 240; erratum, ibid. 3 (1972), 376.
  • Feynman R. P. Atomic theory of the $\lambda$ transition in Helium, Phys. Rev. 91, 1291–1301 (1953)
  • Gandolfo. D, Ruiz. J, Ueltschi. D, On a model of random cycles, J. Statist. Phys. 129, 663–676 (2007) \2360227
  • Hansen, Jennie C. Order statistics for decomposable combinatorial structures. Random Structures Algorithms 5 (1994), no. 4, 517–533.
  • Harris, T. E. Nearest-neighbor Markov interaction processes on multidimensional lattices. Advances in Math. 9, 66–89. (1972).
  • Johnson, Norman L.; Kotz, Samuel; Balakrishnan, N. Discrete multivariate distributions.Wiley Series in Probability and Statistics: Applied Probability and Statistics. A Wiley-Interscience Publication.John Wiley & Sons, Inc., New York, 1997. xxii+299 pp. ISBN: 0-471-12844-9
  • Kerl, John. Shift in critical temperature for random spatial permutations with cycle weights. J. Stat. Phys. 140 (2010), no. 1, 56–75.
  • Lugo, Michael. Profiles of permutations. Electron. J. Combin. 16 (2009), no. 1, Research Paper 99, 20 pp.
  • Matsubara. T, Quantum-statistical theory of liquid Helium, Prog. Theoret. Phys. 6, 714–730 (1951)
  • Schramm, Oded. Compositions of random transpositions. Israel J. Math. 147 (2005), 221–243.
  • Sütó, András. Percolation transition in the Bose gas. J. Phys. A 26 (1993), no. 18, 4689–4710.
  • Sütó, András. Percolation transition in the Bose gas. II. J. Phys. A 35 (2002), no. 33, 6995–7002.
  • Tóth, Bálint. Improved lower bound on the thermodynamic pressure of the spin $1/2$ Heisenberg ferromagnet. Lett. Math. Phys. 28 (1993), no. 1, 75–84.