Limit distributions for Euclidean random permutations

Abstract

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 $d\ge 1$, 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.