Graph limits of random unlabelled $k$-trees

TL;DR

This paper investigates the asymptotic geometric shapes of large random unlabelled $k$-trees using advanced combinatorial and probabilistic methods, providing insights into their global and local limits.

Contribution

It introduces a novel combination of colouring and cycle pointing techniques to analyze the limits of unlabelled $k$-trees, extending previous methods.

Findings

Identification of Gromov-Hausdorff-Prokhorov limits

Description of Benjamini-Schramm local limits

New combinatorial approach for unlabelled $k$-trees

Abstract

We study random unlabelled $k$-dimensional trees by combining the colouring approach by Gainer-Dewar and Gessel (2014) with the cycle pointing method by Bodirsky, Fusy, Kang and Vigerske (2011). Our main applications are Gromov-Hausdorff-Prokhorov and Benjamini-Schramm limits, that describe their asymptotic geometric shape on a global and local scale as the number of hedra tends to infinity.