Treeable Graphings Are Local Limits of Finite Graphs

TL;DR

This paper proves that treeable graphings can be approximated as local limits of finite graphs with bounded degree, extending previous results and supporting the Aldous-Lyons conjecture in this context.

Contribution

It establishes that all treeable graphings are local limits of finite graphs with bounded degree, broadening the class of graphings for which the Aldous-Lyons conjecture holds.

Findings

Treeable graphings are local limits of finite graphs with degree at most d.

Extension of Elek's result from treeings to all treeable graphings.

Supports the validity of the Aldous-Lyons conjecture for a broader class of graphings.

Abstract

Let $\mathbf G$ be a graphing, that is a Borel graph defined by $d$ measure preserving involutions. We prove that if $\mathbf G$ is {\em treeable} then it arises as the local limit of some sequence $(G_n)_{n\in\mathbb{N}}$ of graphs with maximum degree at most $d$. This extends a result by Elek [G. Elek, Note on limits of finite graphs, Combinatorica 27 (2007)] (for $\mathbf G$ a treeing) and consequently extends the domain of the graphings for which Aldous-Lyons conjecture is known to be true.