The Uniform Infinite Cubic Planar Graph

TL;DR

This paper establishes the existence of a new type of infinite cubic planar graph as a local limit of random cubic planar graphs, using complex probabilistic and combinatorial techniques.

Contribution

It introduces the uniform infinite cubic planar graph (UICPG) as a new limit object and describes its construction via random blow-up operations on dual maps.

Findings

The largest 3-connected component contains about 85% of vertices.

Vertices of the largest component concentrate around a linear scale with Airy fluctuations.

The second-largest component is significantly smaller, of order n^{2/3}.

Abstract

We prove that the random simple cubic planar graph $\mathsf{C}_n$ with an even number $n$ of vertices admits a novel uniform infinite cubic planar graph (UICPG) as quenched local limit. We describe how the limit may be constructed by a series of random blow-up operations applied to the dual map of the type~III Uniform Infinite Planar Triangulation established by Angel and Schramm (Comm. Math. Phys., 2003). Our main technical lemma is a contiguity relation between $\mathsf{C}_n$ and a model where the networks inserted at the links of the largest $3$-connected component of $\mathsf{C}_n$ are replaced by independent copies of a specific Boltzmann network. We prove that the number of vertices of the largest $3$-connected component concentrates at $\kappa n$ for $\kappa \approx 0.85085$, with Airy-type fluctuations of order $n^{2/3}$. The second-largest component is shown to have significantly smaller size $O_p(n^{2/3})$.