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)

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.

[error in script] [error in script]

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
Date:	6 June 2011
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
URI:	http://wrap.warwick.ac.uk/id/eprint/40040

Data sourced from Thomson Reuters' Web of Knowledge

Request changes to a record

Actions (login required)

View Item