Limit Distributions for Euclidean Random Permutations

Dor Elboim, Ron Peled*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

9 Scopus citations

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 ρLd 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.

Original languageEnglish
Pages (from-to)457-522
Number of pages66
JournalCommunications in Mathematical Physics
Volume369
Issue number2
DOIs
StatePublished - 1 Jul 2019

Funding

FundersFunder number
ERC Starting
Horizon 2020 Framework Programme678520
Israel Science Foundation861/15

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