The scaling limit of random cubic planar graphs

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 $\gamma >0$.