Limits of random tree-like discrete structures

Abstract

We study a model of random R-enriched trees that is based on weights on the R-structures and allows for a unified treatment of a large family of random discrete structures. We establish novel distributional limits describing local convergence around fixed and random points in this general context, limit theorems for component sizes when R is a composite class, and a Gromov-Hausdorff scaling limit of random metric spaces patched together from independently drawn metrics on the R-structures. Our main applications treat a selection of examples encompassed by this model. We consider random outerplanar maps sampled according to arbitrary weights assigned to their inner faces, and classify in complete generality distributional limits for both the asymptotic local behaviour near the root-edge and near a uniformly at random drawn vertex. We consider random connected graphs drawn according to weights assigned to their blocks and establish a Benjamini-Schramm limit. We also apply our framework to recover in a probabilistic way a central limit theorem for the size of the largest 2-connected component in random graphs from planar-like classes. We prove Benjamini-Schramm convergence of random k-dimensional trees and establish both scaling limits and local weak limits for random planar maps drawn according to Boltzmann-weights assigned to their 2-connected components.

A decorated tree approach to random permutations in substitution-closed classes

“…Among these, the biased Brownian separable permuton µ (p) of parameter p is a random permuton, constructed from a Brownian excursion and independent signs associated with its local minima, see Maazoun [Maa17]. It was proved in [BBF + 17, BBF + 18] that this is a universal limiting object for substitution-closed permutations classes, in the sense that uniform random permutations in many substitution-closed classes converge to µ (p) , for some p. In this article, we give a new proof of this theorem that is based on an extension of Aldous' skeleton decomposition [Ald93] and the framework of random enriched trees and tree-like structures [Stu18b,Stu16].…”

“…A D-decorated (or D-enriched) tree is a rooted locally finite plane tree T , equipped with a function dec : V int (T ) → D from the set of internal vertices of T to D such that the following holds: for each v in V int (T ), the outdegree of v is exactly size(dec(v)). This is a (planar) variant of Labelle's enriched trees [Lab81], which have been studied in [Stu18b,Stu16] from a probabilistic viewpoint.…”

“…The present work takes a probabilistic approach to the analysis of random permutations from substitution-closed classes. We establish a novel encoding of these permutations as decorated conditioned monotype Galton-Watson forests, hence integrating them into the framework of random enriched trees and tree-like structures introduced by Stufler [Stu16,Stu18b]. This yields a unified and powerful way for describing their asymptotic shape on a global and local scale.…”

“…That is, random plane forests where each vertex is enriched with an independent local structure. This integrates the random permutations naturally into the framework of random tree-like structures [Stu16].…”

“…Lemma 4.1 provides a general result to this effect, allowing to condition on the number of vertices with arity in any given set Ω ⊆ N 0 satisfying P(ξ ∈ Ω) > 0. • The literature also contains results on the number of (extended) fringe subtrees of T ξ n (and related models) isomorphic to a given tree [Ald91,Jan12,HJ17,Stu16,Stu19b,Stu19a]. When ξ is critical, such results may be translated to local limit results for T ξ n , pointed at a random leaf (see Proposition 6.13).…”