The GHP Scaling Limit of Uniform Spanning Trees in High Dimensions

TL;DR

This paper shows that a specific mathematical structure called the Brownian continuum random tree emerges as a scaling limit of uniform spanning trees on high-dimensional graphs.

Contribution

The paper establishes the Gromov–Hausdorff–Prohorov scaling limit of uniform spanning trees on high-dimensional graphs.

Findings

The Brownian continuum random tree is the scaling limit of uniform spanning trees on high-dimensional graphs.

The rescaled diameter, height, and random walk on these trees converge to their continuum analogues.

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 \documentclass[12pt]{minimal}
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\begin{document}$$\{0,1\}^n$$\end{document}{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.