Grosskinsky, Stefan, Lovisolo, Alexander A. and Ueltschi, Daniel, 1969-. (2012) Lattice permutations and Poisson-Dirichlet distribution of cycle lengths. Journal of Statistical Physics, Vol.146 (No.6). pp. 1105-1121. ISSN 0022-4715

PDF WRAP_Grosskinsky_Lattice_1107.5215v2.pdf - Accepted Version Restricted to Repository staff only until 2 March 2013. - Requires a PDF viewer such as GSview, Xpdf or Adobe Acrobat Reader Download (780Kb)

Abstract

We study random spatial permutations on ℤ3 where each jump x↦π(x) is penalized by a factor e−T∥x−π(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.

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:	Journal of Statistical Physics
Publisher:	Springer
ISSN:	0022-4715
Date:	March 2012
Volume:	Vol.146
Number:	No.6
Page Range:	pp. 1105-1121
Identification Number:	10.1007/s10955-012-0450-9
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Restricted or Subscription Access
Funder:	Erasmus Mundus (Program), Engineering and Physical Sciences Research Council (EPSRC)
Grant number:	EP/E501311/1 (EPSRC), EP/G056390/1 (EPSRC)
References:	[1] M. Aizenman, Geometric analysis of '4 Fields and Ising models, Commun. Math. Phys. 86, 1–48 (1982) [2] M. Aizenman, B. Nachtergaele, Geometric aspects of quantum spin states, Comm. Math. Phys. 164, 17–63 (1994) [3] D. Aldous, Deterministic and stochastic models for coalescence (aggregation and coagula- tion): a review of the mean-field theory for probabilists, Bernoulli 5, 3–48 (1999) [4] R. Arratia, A. D. Barbour, S. Tavar´e, Logarithmic Combinatorial Structures: A Probabilistic Approach. EMS Monographs in Mathematics (2003) [5] N. Berestycki, Emergence of giant cycles and slowdown transition in random transpositions and k-cycles, Electr. J. Probab. 16, 152–173 (2011) [6] N. Berestycki, R. Durrett Limiting behavior for the distance of a random walk, Electr. J. Probab. 13, 374–395 (2008) [7] J. Bertoin, Random Fragmentation and Coagulation Processes. Cambridge University Press (2006) [8] V. Betz, D. Ueltschi, Spatial random permutations and infinite cycles, Commun. Math. Phys. 285, 469–501 (2009) [9] V. Betz, D. Ueltschi, Spatial random permutations and Poisson-Dirichlet law of cycle lengths, Electr. J. Probab. 16, 1173-1192 (2011) [10] N. Crawford, D. Ioffe, Random current representation for transverse field Ising model, Commun. Math. Phys. 296, 447–474 (2010) [11] P. Diaconis, E. Mayer-Wolf, O. Zeitouni, M. P. W. Zerner, The Poisson-Dirichlet law is the unique invariant distribution for uniform split-merge transformations, Ann. Probab. 32, 915–938 (2004) [12] S. Feng, The Poisson-Dirichlet Distribution and Related Topics. Probability and its Applications, Springer (2010) [13] R. P. Feynman, Atomic theory of the � transition in Helium, Phys. Rev. 91, 1291–1301 (1953) [14] G. Grimmett, Space-time percolation, In “In and out of equilibrium. 2”, Progr. Probab. 60, 305–320 (2008) [15] D. Gandolfo, J. Ruiz, D. Ueltschi, On a model of random cycles, J. Statist. Phys. 129, 663–676 (2007) [16] C. Goldschmidt, D. Ueltschi, P. Windridge, Quantum Heisenberg models and their prob- abilistic representations, Entropy and the Quantum II, Contemporary Mathematics 552, 177–224 (2011); arXiv:1104.0983 [17] J. Kerl, Shift in critical temperature for random spatial permutations with cycle weights, J. Statist. Phys. 140, 56–75 (2010) [18] J. F. C. Kingman, Random discrete distributions, J. Roy. Statist. Soc. B 37, 1–15 (1975) [19] J. F. C. Kingman, Mathematics of genetic diversity. CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 34, SIAM (1980) [20] J. Pitman, M. Yor, The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator, Ann. Probab. 25, 855–900 (1997) [21] J. Pitman, Poisson-Dirichlet and GEM invariant distributions for split-and-merge trans- formations of an interval partition, Combinatorics, Probability and Computing (2002), 11: 501-514 [22] O. Schramm, Compositions of random transpositions, Israel J. Math. 147, 221–243 (2005) [23] L. A. Shepp, S. L. Lloyd, Ordered cycle lengths in a random permutation, Trans. Amer. Math. Soc. 121, 340–357 (1966) [24] A. S¨ut˝o, Percolation transition in the Bose gas, J. Phys. A 26, 4689–4710 (1993) [25] B. T´oth, Improved lower bound on the thermodynamic pressure of the spin 1/2 Heisenberg ferromagnet, Lett. Math. Phys. 28, 75 (1993) [26] N. V. Tsilevich, Stationary random partitions of a natural series, Teor. Veroyatnost. i Primenen. 44, 55–73 (1999) [27] D. Wilson, Mixing times of lozenge tiling and card shuffling Markov chains, Ann. Appl. Probab. 14, 274–325 (2004)
URI:	http://wrap.warwick.ac.uk/id/eprint/44525

Request changes to a record

Actions (login required)

View Item