The scaling limit of random cubic planar graphs

Abstract

We study the random cubic planar graph $C_n$ with an even number $n$ of vertices. We show that the Brownian map arises as Gromov–Hausdorff–Prokhorov scaling limit of $C_n$ as $n\in 2N$ tends to infinity, after rescaling distances by $\gamma n^{-1/4}$ for a specific constant $\gamma >0$. This is the first time a model of random graphs that are not embedded into the plane is shown to converge to the Brownian map. Our approach features a new method that allows us to relate distances on random graphs to first-passage percolation distances on their 3-connected core.