Thresholds for geometric graphs

TL;DR

This paper investigates the conditions under which random geometric graphs on various metric spaces exhibit threshold phenomena for all monotone properties, linking thresholds to uniform expansion.

Contribution

It establishes a connection between threshold existence and uniform expansion, proving that standard tori, spheres, and cubes admit thresholds for all monotone properties.

Findings

Standard tori, spheres, and cubes admit thresholds for all monotone properties.

Threshold existence is linked to the uniform expansion of the metric space.

The paper provides a theoretical framework connecting geometric properties to threshold phenomena.

Abstract

A metric probability space $M$ admits thresholds if the random geometric graph on $M$ has a threshold for every monotone graph property. We connect the existence of thresholds to the uniform expansion of $M$ and prove that all standard tori, spheres, and cubes admit thresholds.