Limits of random tree-like discrete structures

TL;DR

This paper develops a unified probabilistic framework for analyzing the local and global limits of various random discrete structures, including trees, maps, and graphs, using enriched tree models and metric space convergence.

Contribution

It introduces a general model of weighted $ $-enriched trees that captures many random structures and derives their distributional and scaling limits.

Findings

Established local convergence for random $ $-enriched trees.

Derived Gromov--Hausdorff scaling limits for metric spaces from these trees.

Classified distributional limits for random outerplanar maps and block-weighted graphs.

Abstract

We study a model of random $\mathcal{R}$-enriched trees that is based on weights on the $\mathcal{R}$-structures and allows for a unified treatment of a large family of random discrete structures. We establish distributional limits describing local convergence around fixed and random points in this general context, limit theorems for component sizes when $\mathcal{R}$ is a composite class, and a Gromov--Hausdorff scaling limit of random metric spaces patched together from independently drawn metrics on the $\mathcal{R}$-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.