Local limit of sparse random planar graphs

Abstract

Let $P(n,m)$ be a graph chosen uniformly at random from the class of all planar graphs on vertex set $\left\{1,\dots ,n\right\}$ with $m=m(n)$ edges. We determine the (Benjamini-Schramm) local weak limit of $P(n,m)$ in the sparse regime when $m\le n+o\left(n{\left(\log n\right)}^{-2/3}\right)$. Assuming that the average degree $2m/n$ tends to a constant $c\in [0,2]$ the local weak limit of $P(n,m)$ is a Galton-Watson tree with offspring distribution $Po(c)$ if $c\le 1$, while it is the Skeleton tree if $c=2$. Furthermore, there is a smooth transition between these two cases in the sense that the local weak limit of $P(n,m)$ is a linear combination of a Galton-Watson tree and the Skeleton tree if $c\in \left(1,2\right)$.