Limit Distributions for Euclidean Random Permutations

We study the length of cycles in the model of spatial random permutations in Euclidean space. In this model, for given length L, density ρ, dimension d and jump density φ, one samples ρL^d particles in a d-dimensional torus of side length L, and a permutation π of the particles, with probability density proportional to the product of values of φ at the differences between a particle and its image under π. The distribution may be further weighted by a factor of θ to the number of cycles in π. Following Matsubara and Feynman, the emergence of macroscopic cycles in π at high density ρ has been related to the phenomenon of Bose–Einstein condensation. For each dimension d≥ 1 , we identify sub-critical, critical and super-critical regimes for ρ and find the limiting distribution of cycle lengths in these regimes. The results extend the work of Betz and Ueltschi. Our main technical tools are saddle-point and singularity analysis of suitable generating functions following the analysis by Bogachev and Zeindler of a related surrogate-spatial model.