The scaling limit of random cubic planar graphs

Abstract

We study the random cubic planar graph 𝖢 𝑛 with aneven number 𝑛 of vertices. We show that the Brown-ian map arises as Gromov–Hausdorff–Prokhorov scalinglimit of 𝖢 𝑛 as 𝑛 ∈ 2 ℕ tends to infinity, after rescaling dis-tances by 𝛾𝑛⁻¹∕⁴ for a specific constant 𝛾 > 0. This is thefirst time a model of random graphs that are not embed-ded into the plane is shown to converge to the Brownianmap. Our approach features a new method that allowsus to relate distances on random graphs to first-passagepercolation distances on their 3-connected core.