The GHP scaling limit of uniform spanning trees in high dimensions

TL;DR

This paper proves that in high dimensions, the uniform spanning tree converges to the Brownian continuum random tree when scaled appropriately, revealing universal geometric and probabilistic properties.

Contribution

It establishes the Gromov-Hausdorff-Prohorov scaling limit of uniform spanning trees in high-dimensional graphs, extending known results to new graph classes.

Findings

Convergence of rescaled diameter, height, and random walk to continuum analogues

Universal scaling limit for uniform spanning trees in high dimensions

Application to various high-dimensional graph models

Abstract

We show that the Brownian continuum random tree is the Gromov-Hausdorff-Prohorov scaling limit of the uniform spanning tree on high-dimensional graphs including the $d$-dimensional torus $\mathbb{Z}_n^d$ with $d>4$, the hypercube $\{0,1\}^n$, and transitive expander graphs. Several corollaries for associated quantities are then deduced: convergence in distribution of the rescaled diameter, height and simple random walk on these uniform spanning trees to their continuum analogues on the continuum random tree.