Random cubic planar graphs converge to the Brownian sphere

TL;DR

This paper proves that the large-scale structure of random cubic planar graphs converges to the Brownian sphere, using decomposition into 3-connected components and duality with triangulations.

Contribution

It establishes the scaling limit of random cubic planar graphs as the Brownian sphere by extending existing frameworks to these graphs and their dual triangulations.

Findings

Random cubic planar graphs converge to the Brownian sphere.

Decomposition into 3-connected components approximates the metric structure.

Duality with triangulations facilitates the convergence proof.

Abstract

In this paper, the scaling limit of random connected cubic planar graphs (respectively multigraphs) is shown to be the Brownian sphere. The proof consists in essentially two main steps. First, thanks to the known decomposition of cubic planar graphs into their 3-connected components, the metric structure of a random cubic planar graph is shown to be well approximated by its unique 3-connected component of linear size, with modified distances. Then, Whitney's theorem ensures that a 3-connected cubic planar graph is the dual of a simple triangulation, for which it is known that the scaling limit is the Brownian sphere. Curien and Le Gall have recently developed a framework to study the modification of distances in general triangulations and in their dual. By extending this framework to simple triangulations, it is shown that 3-connected cubic planar graphs with modified distances converge jointly with their dual triangulation to the Brownian sphere.