Spatial Random Permutations and Poisson-Dirichlet Law of Cycle Lengths

Volker Betz (University of Warwick)
Daniel Ueltschi (University of Warwick)

Abstract


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.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1173-1192

Publication Date: June 6, 2011

DOI: 10.1214/EJP.v16-901

References

  1. Arratia, Richard; Barbour, A. D.; Tavaré, Simon. Random combinatorial structures and prime factorizations. Notices Amer. Math. Soc. 44 (1997), no. 8, 903--910. MR1467654 (98i:60007)
  2. Berestycki. N, Emergence of giant cycles and slowdown transition in random transpositions and $k$-cycles, Electr. J. Probab. 16, 152--173 (2011)
  3. Betz, Volker; Ueltschi, Daniel. Spatial random permutations and infinite cycles. Comm. Math. Phys. 285 (2009), no. 2, 469--501. MR2461985 (2009h:82045)
  4. Betz. V, Ueltschi. D. Spatial random permutations with small cycle weights}, Probab. Th. Rel. Fields 149, 191--222 (2011)
  5. Betz. V, Ueltschi. D, Critical temperature of dilute Bose gases, Phys. Rev. A 81, 023611 (2010)
  6. Betz, Volker; Ueltschi, Daniel; Velenik, Yvan. Random permutations with cycle weights. Ann. Appl. Probab. 21 (2011), no. 1, 312--331. MR2759204
  7. Biskup. M, Richthammer. T. in preparation
  8. Buffet, E.; Pulé, J. V. Fluctuation properties of the imperfect Bose gas. J. Math. Phys. 24 (1983), no. 6, 1608--1616. MR0708683 (85d:82011)
  9. 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. MR0325177 (48 #3526)
  10. Feynman R. P. Atomic theory of the $\lambda$ transition in Helium, Phys. Rev. 91, 1291--1301 (1953)
  11. Gandolfo. D, Ruiz. J, Ueltschi. D, On a model of random cycles, J. Statist. Phys. 129, 663--676 (2007) \MR{2360227}
  12. Hansen, Jennie C. Order statistics for decomposable combinatorial structures. Random Structures Algorithms 5 (1994), no. 4, 517--533. MR1293077 (96f:60010)
  13. Harris, T. E. Nearest-neighbor Markov interaction processes on multidimensional lattices. Advances in Math. 9, 66--89. (1972). MR0307392 (46 #6512)
  14. 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 MR1429617 (99h:62014)
  15. Kerl, John. Shift in critical temperature for random spatial permutations with cycle weights. J. Stat. Phys. 140 (2010), no. 1, 56--75. MR2651438 (2011g:82038)
  16. Lugo, Michael. Profiles of permutations. Electron. J. Combin. 16 (2009), no. 1, Research Paper 99, 20 pp. MR2529808 (2010k:05025)
  17. Matsubara. T, Quantum-statistical theory of liquid Helium, Prog. Theoret. Phys. 6, 714--730 (1951)
  18. Schramm, Oded. Compositions of random transpositions. Israel J. Math. 147 (2005), 221--243. MR2166362 (2006h:60024)
  19. Sütő, András. Percolation transition in the Bose gas. J. Phys. A 26 (1993), no. 18, 4689--4710. MR1241339 (94g:82005)
  20. Sütő, András. Percolation transition in the Bose gas. II. J. Phys. A 35 (2002), no. 33, 6995--7002. MR1945163 (2003k:82012)
  21. 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. MR1224836 (94h:82062)
Bookmark and Share


Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.