Random permutations of a regular lattice

Abstract

Spatial random permutations were originally studied due to their connections to Bose-Einstein condensation, but they possess many interesting properties of their own. For random permutations of a regular lattice with periodic boundary conditions, we prove existence of the infinite volume limit under fairly weak assumptions. When the dimension of the lattice is two, we give numerical evidence of a Kosterlitz-Thouless transition, and of long cycles having an almost sure fractal dimension in the scaling limit. Finally we comment on possible connections to Schramm-Löwner curves.

23 pages, 8 figures

References in corpus (5)

• Spatial random permutations and infinite cycles
• Spatial random permutations and Poisson-Dirichlet law of cycle lengths
• Lattice permutations and Poisson-Dirichlet distribution of cycle lengths
• The relation between Feynman cycles and off-diagonal long-range order
• Shift in critical temperature for random spatial permutations with cycle weights

Cited by in corpus (3)

• Site monotonicity and uniform positivity for interacting random walks and the spin O(N) model with arbitrary N
• Loop correlations in random wire models
• Gibbs measures on permutations over one-dimensional discrete point sets