Lattice Permutations and Poisson-Dirichlet Distribution of Cycle Lengths

We study random spatial permutations on ℤ³ where each jump x↦π(x) is penalized by a factor e^{-T|| x-p(x)||2}\mathrm{e}^{-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.