Lattice Permutations and PoissonDirichlet Distribution of Cycle Lengths
Abstract
We study random spatial permutations on &Z;^{3} where each jump x↦ π( x) is penalized by a factor e^{T\ xπ (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 PoissonDirichlet. This can be explained heuristically using a stochastic coagulationfragmentation process for long cycles, which is supported by numerical data.
 Publication:

Journal of Statistical Physics
 Pub Date:
 March 2012
 DOI:
 10.1007/s1095501204509
 arXiv:
 arXiv:1107.5215
 Bibcode:
 2012JSP...146.1105G
 Keywords:

 Lattice permutations;
 Cycle lengths;
 PoissonDirichlet distribution;
 Condensed Matter  Statistical Mechanics;
 Mathematical Physics;
 60K35;
 82B20
 EPrint:
 18 pages, 14 figures