Lattice Permutations and Poisson-Dirichlet Distribution of Cycle Lengths

@article{Grosskinsky2012LatticePA,
  title={Lattice Permutations and Poisson-Dirichlet Distribution of Cycle Lengths},
  author={Stefan Grosskinsky and Alexander A. Lovisolo and Daniel Ueltschi},
  journal={Journal of Statistical Physics},
  year={2012},
  volume={146},
  pages={1105-1121}
}
We study random spatial permutations on ℤ3 where each jump x↦π(x) is penalized by a factor $\mathrm{e}^{-T\| x-\pi (x)\|^{2}}$. 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. 

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