Geometric random graphs on circles

TL;DR

This paper investigates infinite random geometric graphs on circles, revealing that the graph's structure almost surely depends on whether the circle's circumference is rational or irrational, extending previous work beyond Banach spaces.

Contribution

It extends the study of random geometric graphs to circular metric spaces, showing a surprising dependence on the rationality of the circle's circumference.

Findings

Graph structure depends on the rationality of the circle's circumference.

Almost sure isomorphism to a fixed graph varies with rationality.

Extends previous work from Banach spaces to circular spaces.

Abstract

Given a dense countable set in a metric space, the infinite random geometric graph is the random graph with the given vertex set and where any two points at distance less than 1 are connected, independently, with some fixed probability. It has been observed by Bonato and Janssen that in some, but not all, such settings, the resulting graph does not depend on the random choices, in the sense that it is almost surely isomorphic to a fixed graph. While this notion makes sense in the general context of metric spaces, previous work has been restricted to sets in Banach spaces. We study the case when the underlying metric space is a circle of circumference $L$, and find a surprising dependency of behavior on the rationality of $L$.