Local limit of sparse random planar graphs

TL;DR

This paper characterizes the local weak limit of sparse random planar graphs, showing a transition from Galton-Watson trees to Skeleton trees as the average degree varies from 0 to 2.

Contribution

It determines the local weak limit of sparse random planar graphs in the regime where the number of edges is close to the number of vertices, revealing a phase transition in the limiting structure.

Findings

For average degree c ≤ 1, the limit is a Galton-Watson tree with Po(c) offspring.

At c=2, the limit is the Skeleton tree.

For c in (1,2), the limit is a mixture of Galton-Watson and Skeleton trees.

Abstract

Let $P(n,m)$ be a graph chosen uniformly at random from the class of all planar graphs on vertex set $\left\{1, \ldots, n\right\}$ with $m=m(n)$ edges. We determine the (Benjamini-Schramm) local weak limit of $P(n,m)$ in the sparse regime when $m\leq n+o\left(n\left(\log n\right)^{-2/3}\right)$. 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 $c\leq 1$, 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 $c\in\left(1,2\right)$.