Random enriched trees with applications to random graphs

TL;DR

This paper develops limit theorems for random enriched trees considering symmetry, and applies these to analyze the asymptotic geometric properties of various classes of unlabelled random graphs and k-trees.

Contribution

The paper introduces general limit theorems for symmetric random enriched trees and applies them to unlabelled graphs and k-trees, establishing their scaling and local limits.

Findings

Gromov--Hausdorff scaling limits for models

Benjamini--Schramm limits identified

Local weak limits characterized near the root

Abstract

We establish limit theorems that describe the asymptotic local and global geometric behaviour of random enriched trees considered up to symmetry. We apply these general results to random unlabelled weighted rooted graphs and uniform random unlabelled $k$-trees that are rooted at a $k$-clique of distinguishable vertices. For both models we establish a Gromov--Hausdorff scaling limit, a Benjamini--Schramm limit, and a local weak limit that describes the asymptotic shape near the fixed root.