Limit distributions for Euclidean random permutations

TL;DR

This paper investigates the limiting behavior of cycle lengths in Euclidean random permutations, revealing phase transitions and distribution regimes related to Bose-Einstein condensation across different dimensions.

Contribution

It extends previous work by identifying sub-critical, critical, and super-critical regimes for cycle lengths in Euclidean random permutations, using advanced analytical techniques.

Findings

Identifies regimes for cycle length distributions in various dimensions.

Establishes connections between cycle structure and Bose-Einstein condensation.

Provides explicit limiting distributions in different density regimes.

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