The uniform infinite cubic planar graph

Abstract

We prove that the random simple connected cubic planar graph Cn 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. 241 (2003) 191–213). Our main technical lemma is a contiguity relation between Cn and a model where the networks inserted at the links of the largest 3-connected component of Cn 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 κn for κ ≈ 0.85085, with Airy-type fluctuations of order n2/3. The second-largest component is shown to have significantly smaller size OP n2/3.