Spatial random permutations and infinite cycles

TL;DR

This paper studies spatial random permutations, revealing the conditions for infinite cycles at high densities and connecting the model to quantum Bose gases, with an exact critical density expression.

Contribution

It provides an exact expression for the critical density and explores the relationship between spatial permutations and quantum Bose gases.

Findings

Infinite cycles occur at high densities in spatial permutations.

An exact formula for the critical density is derived.

The model relates to both ideal and interacting quantum Bose gases.

Abstract

We consider systems of spatial random permutations, where permutations are weighed according to the point locations. Infinite cycles are present at high densities. The critical density is given by an exact expression. We discuss the relation between the model of spatial permutations and the ideal and interacting quantum Bose gas.