Lattice Permutations and Poisson-Dirichlet Distribution of Cycle Lengths

We study random spatial permutations on acurrency sign(3) where each jump xa dagger broken vertical bar pi(x) is penalized by a factor 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.