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 $\ddot{I}$, dimension $d$ and jump density $\ddot{I}†$, one samples $\ddot{I}L^d$ particles in a $d$-dimensional torus of side length $L$, and a permutation $\ddot{I}€$ of the particles, with probability density proportional to the product of values of $\ddot{I}†$ at the differences between a particle and its image under $\ddot{I}€$. The distribution may be further weighted by a factor of $\hat{I}¸$ to the number of cycles in $\ddot{I}€$. Following Matsubara and Feynman, the emergence of macroscopic cycles in $\ddot{I}€$ at high density $\ddot{I}$ 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 $\ddot{I}$ 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.

Fixed broken citation link. 58 pages, 6 figures