# 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.

## Key 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.