Lattice permutations and Poisson-Dirichlet distribution of cycle lengths

We study random spatial permutations on Z^3 where each jump x -> π(x) is penalized by a factor exp(-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.

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