The scaling limit of random cubic planar graphs

TL;DR

This paper proves that large random cubic planar graphs, when scaled appropriately, converge to the Brownian map in the Gromov--Hausdorff--Prokhorov sense, revealing their universal geometric limit.

Contribution

It establishes the scaling limit of random cubic planar graphs as the Brownian map, a significant step in understanding their geometric structure.

Findings

Convergence of scaled cubic planar graphs to the Brownian map

Identification of the scaling factor b3 n^{-1/4}

Extension of known limits from other graph classes

Abstract

We study the random simple connected cubic planar graph $\mathsf{C}_n$ with an even number $n$ of vertices. We show that the Brownian map arises as Gromov--Hausdorff--Prokhorov scaling limit of $\mathsf{C}_n$ as $n \in 2 \ndN$ tends to infinity, after rescaling distances by $\gamma n^{-1/4} $ for a specific constant $\gamma>0$.