The GHP scaling limit of uniform spanning trees of dense graphs

TL;DR

This paper proves that uniform spanning trees of dense graph sequences converge to the Brownian CRT under GHP scaling, with explicit constants derived from the graphon, applicable to Erdős-Rényi graphs and dense expanders.

Contribution

It establishes the GHP scaling limit of uniform spanning trees for dense graph sequences converging to a graphon, including explicit scaling constants.

Findings

Uniform spanning trees of dense graphs converge to the Brownian CRT.

Explicit scaling constants are derived from the limiting graphon.

Convergence implies the convergence of height, diameter, and random walk laws.

Abstract

We consider dense graph sequences that converge to a connected graphon and prove that the GHP scaling limit of their uniform spanning trees is Aldous' Brownian CRT. Furthermore, we are able to extract the precise scaling constant from the limiting graphon. As an example, we can apply this to the scaling limit of the uniform spanning trees of the Erd\"os-R\'enyi sequence $(G(n,p))_{n \geq 1}$ for any fixed $p \in (0,1]$, and sequences of dense expanders. A consequence of GHP convergence is that several associated quantities of the spanning trees also converge, such as the height, diameter and law of a simple random walk.