Random permutations of a regular lattice

TL;DR

This paper investigates the properties of spatial random permutations on regular lattices, proving the existence of an infinite volume limit, providing evidence of a phase transition in two dimensions, and exploring connections to fractal geometry and Schramm-L"owner curves.

Contribution

It establishes the existence of the infinite volume limit for lattice permutations and provides numerical evidence of a Kosterlitz-Thouless transition in two dimensions.

Findings

Existence of infinite volume limit under weak assumptions.

Numerical evidence of a Kosterlitz-Thouless transition in 2D.

Long cycles exhibit fractal dimensions in the scaling limit.

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-L\"owner curves.