Local convergence of random planar graphs

Abstract

The present work describes the asymptotic local shape of a graph drawn uniformly at random from all connected simple planar graphs with $n$ labelled vertices. We establish a novel uniform infinite planar graph (UIPG) as quenched limit in the local topology as $n\to \infty$. We also establish such limits for random $2$-connected planar graphs and maps as their number of edges tends to infinity. Our approach encompasses a new probabilistic view on the Tutte decomposition. This allows us to follow the path along the decomposition of connectivity from planar maps to planar graphs in a uniform way, basing each step on condensation phenomena for random walks under subexponentiality and Gibbs partitions. Using large deviation results, we recover the asymptotic formula by Giménez and Noy (2009) for the number of planar graphs.