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 , dimension d and jump density phi, one samples Ld particles in a d-dimensional torus of side length L, and a permutation of the particles, with probability density proportional to the product of values of phi at the differences between a particle and its image under . The distribution may be further weighted by a factor of to the number of cycles in . Following Matsubara and Feynman, the emergence of macroscopic cycles in at high density has been related to the phenomenon of Bose-Einstein condensation. For each dimension d1, we identify sub-critical, critical and super-critical regimes for 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.