TY - JOUR

T1 - Limit Distributions for Euclidean Random Permutations

AU - Elboim, Dor

AU - Peled, Ron

N1 - Publisher Copyright:
© 2019, Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2019/7/1

Y1 - 2019/7/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85064903255&partnerID=8YFLogxK

U2 - 10.1007/s00220-019-03421-8

DO - 10.1007/s00220-019-03421-8

M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???

AN - SCOPUS:85064903255

SN - 0010-3616

VL - 369

SP - 457

EP - 522

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

IS - 2

ER -