The Library

Spatial random permutations and Poisson-Dirichlet law of cycle lengths

Tools

Betz, Volker and Ueltschi, Daniel. (2011) Spatial random permutations and Poisson-Dirichlet law of cycle lengths. Electronic Journal of Probability, Vol.16 (No.41). ISSN 1083-6489

Preview

PDF
WRAP_Betz_Spatial_random_permutations.pdf - Published Version - Requires a PDF viewer such as GSview, Xpdf or Adobe Acrobat Reader
Download (323Kb)

Official URL: http://128.208.128.142/~ejpecp/viewarticle.php?id=...

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.

Item Type:

Journal Article

Subjects:

Q Science > QA Mathematics

Divisions:

Faculty of Science > Mathematics

Library of Congress Subject Headings (LCSH):

Permutations

Journal or Publication Title:

Electronic Journal of Probability

Publisher:

University of Washington. Dept. of Mathematics

ISSN:

1083-6489

Official Date:

6 June 2011

Dates:

Date	Event
6 June 2011	Published

Volume:

Vol.16

Number:

No.41

Status:

Peer Reviewed

Publication Status:

Published

Access rights to Published version:

Open Access

Funder:

Engineering and Physical Sciences Research Council (EPSRC)

Grant number:

EP/D07181X/1 (EPSRC), EP/G056390/1 (EPSRC)

Related URLs:

Publisher

References:

[1] R. Arratia, A. D. Barbour, S. Tavaré, Random combinatorial structures and prime factorizations,
Notices Amer. Math. Soc. 44, 903–910 (1997) MR1467654
[2] N. Berestycki, Emergence of giant cycles and slowdown transition in random transpositions and
k-cycles, Electr. J. Probab. 16, 152–173 (2011)
[3] V. Betz, D. Ueltschi, Spatial random permutations and infinite cycles, Commun. Math. Phys.
285, 469–501 (2009) MR2461985
[4] V. Betz, D. Ueltschi, Spatial random permutations with small cycle weights, Probab. Th. Rel.
Fields 149, 191–222 (2011)
[5] V. Betz, D. Ueltschi, Critical temperature of dilute Bose gases, Phys. Rev. A 81, 023611 (2010)
[6] V. Betz, D. Ueltschi, Y. Velenik, Random permutations with cycle weights, Ann. Appl. Prob. 21,
312–331 (2011) MR2759204
[7] M. Biskup, T. Richthammer, in preparation
[8] E. Buffet, J. V. Pulé, Fluctuation properties of the imperfect Bose gas, J. Math. Phys. 24, 1608–
1616 (1983) MR0708683
[9] W. J. Ewens, The sampling theory of selectively neutral alleles, Theoret. Populations Bil. 3, 87–
112 (1972) MR0325177
[10] R. P. Feynman, Atomic theory of the � transition in Helium, Phys. Rev. 91, 1291–1301 (1953)
[11] D. Gandolfo, J. Ruiz, D. Ueltschi, On a model of random cycles, J. Statist. Phys. 129, 663–676
(2007) MR2360227
[12] J. C. Hansen, Order statistics for decomposable combinatorial structures, Random Structures
Algorithms 5, 517–533 (1994) MR1293077
[13] T. E. Harris, Nearest neighbour Markov interaction processes on multidimensional lattices, Adv.
Math. 9, 66 (1972) MR0307392
[14] N. L. Johnson, S. Kotz, N. Balakrishnan, Discrete Multivariate Distributions, John Wiley & Sons
(1997) MR1429617
[15] J. Kerl, Shift in critical temperature for random spatial permutations with cycle weights, J. Statist.
Phys. 140, 56–75 (2010) MR2651438
[16] M. Lugo, Profiles of permutations, Electr. J. Comb. 16, R99 (2009) MR2529808
[17] T. Matsubara, Quantum-statistical theory of liquid Helium, Prog. Theoret. Phys. 6, 714–730
(1951)
[18] O. Schramm, Compositions of random transpositions, Israel J. Math. 147, 221–243 (2005)
MR2166362
[19] A. Süt˝o, Percolation transition in the Bose gas, J. Phys. A 26, 4689–4710 (1993) MR1241339
[20] A. Süt˝o, Percolation transition in the Bose gas II, J. Phys. A 35, 6995–7002 (2002) MR1945163
[21] B. Tóth, Improved lower bound on the thermodynamic pressure of the spin 1=2 Heisenberg ferromagnet,
Lett. Math. Phys. 28, 75 (1993) MR1224836