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 {1,... ,n} with m = m(n) edges. We determine the (Benjamini-Schramm) local weak limit of P(n,m) in the sparse regime when m ≤ n + o n log n −2/3 . Assuming that the average degree 2m/n tends to a constant c ∈ [0,2] the local weak limit of P(n,m) is a Galton-Watson tree with offspring distribution Po (c) if c ≤ 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 ∈ (1,2). d − − → GWT (c).

The Uniform Infinite Cubic Planar Graph

“…The limit UICPG is almost surely recurrent by the famous result [5] for locally convergent sequences of random graphs with bounded degrees, which was later generalized in [17] to graphs with light-tailed degree distributions. The name Uniform Infinite Planar Cubic Graph follows the naming tradition of limits for different models of random networks, as in the pioneering work on the Uniform Infinite Planar Triangulation [2], and further work in this active research field [7,28,9,11,10,22,30,20]. It appears that still less is known about cubic planar graphs than about these models.…”