Local convergence of random planar graphs (Q2693140)

Summary: 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 \textit{O. Giménez} and \textit{M. Noy} [J. Am. Math. Soc. 22, No. 2, 309--329 (2009; Zbl 1206.05019)] for the number of planar graphs.