Phase transition in random distance graphs on the torus

TL;DR

This paper studies phase transitions in random distance graphs on the torus, revealing they can behave like Erdős-Rényi models in both supercritical and subcritical phases, expanding understanding of solvable inhomogeneous random graphs.

Contribution

It introduces a new class of solvable inhomogeneous random graphs outside rank 1 models, demonstrating their phase transition behavior matches Erdős-Rényi models.

Findings

Random distance graphs on the torus exhibit Erdős-Rényi-like phase transitions.

The models are solvable for the largest connected component analysis.

Behavior is consistent in both supercritical and subcritical regimes.

Abstract

We apply here methods of inhomogeneous random graphs to a class of random distance graphs. This provides an example outside of the rank 1 models which is still solvable as long as the largest connected component is concerned. In particular, we show that some random distance graphs behave exactly as the classical Erd\H{o}s-R\'enyi model not only in the supercritical phase (as was already known) but in the subcritical case as well.