# 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.

## 31 Citations

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