## Lattice permutations and Poisson-Dirichlet distribution of cycle lengths    [PDF]

Stefan Grosskinsky, Alexander A. Lovisolo, Daniel Ueltschi
We study random spatial permutations on Z^3 where each jump x -> \pi(x) is penalized by a factor exp(-T ||x-\pi(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.
View original: http://arxiv.org/abs/1107.5215