Spatial Random Permutations and Poisson-Dirichlet Law of Cycle Lengths
Volker Betz, Daniel Ueltschi · Electronic Journal of Probability · 2011
We study spatial permutations with cycle weights that are bounded or slowly diverging. We show that a phase transition occurs at an explicit critical density. The long cycles are macroscopic and their cycle lengths satisfy a Poisson-Dirichlet law.