Scaling limits of random graphs from subcritical classes

Abstract

Abstract. We study the uniform random graph Cn with n vertices drawn from a subcritical class of connected graphs. Our main result is that the rescaled graph Cn/ √ n converges to the Brownian Continuum Random Tree Te multiplied by a constant scaling factor that depends on the class under consideration. In addition, we provide subgaussian tail bounds for the diameter D(Cn) and height H(C • n ) of the rooted random graph C • n . We give analytic expressions for the scaling factor of several classes, including for example the prominent class of outerplanar graphs. Our methods also enable us to study first passage percolation on Cn, where we show the convergence to Te under an appropriate rescaling.

Scaling limits of random outerplanar maps with independent link-weights

“…More generally, we may define the constant µ(y) for arbitrary parameters y with 0 < C out * (y) < ∞. This constant was computed in [PSW14] in order to compute the scaling constant of outerplanar graphs. …”

Scaling limit of random k-trees

“…The concept Continuum Random Tree was also introduced by Aldous [2,3,4] and further developed by Duquesne and Le Gall [19,20,21]. Since Aldous's pioneering work on the GaltonWatson trees, the CRT has been established as the limiting object of a large variety of combinatorial structures [30,41,38,39,12,33,8,13,36,10]. A key idea in the study of these combinatorial objects is to relate them to trees endowed with additional structures by using an appropriate bijection.…”

“…Furthermore, in order to build a connection between the distance of two vertices in a random k-tree and the distance of two vertices in a critical Galton-Watson tree, we need to introduce the concept of a size-biased enriched tree. This is adapted from the size-biased Galton-Watson tree which was introduced by Lyons, Pemantle and Peres [35] and was used by Addario-Berry, Devroye and Janson in [1], and was further generalized to the size-biased R-enriched trees by Panagiotou, Stufler and Weller in [38,39]. Our enriched tree is slightly different to the size-biased Renriched tree and we use their ideas in [41,38] where an important step is to relate the distance between two vertices in a random graph to the distance between two blocks in a random size-biased R-enriched tree.…”

“…For this purpose we consider the spine of a size-biased enriched tree, which was adapted from the size-biased Galton-Watson tree in [1]. This idea has been used in studying the scaling limit of random graphs from subcritical graph classes [38] and was further generalized to the random R-enriched trees [42].…”

Asymptotic Enumeration of Graph Classes with Many Components

“…If a block-stable class of connected graphs C additionally satisfies that ζ ∈ (0, R), then C is termed subcritical. Equivalently, it is shown in [15] (Lemma 3.8) that C is subcritical if and only if RB (R) > 1. In this case ζB (ζ) = 1.…”