Scaling limits of the uniform spanning tree and loop-erased random walk on finite graphs

TL;DR

This paper investigates the scaling limits of loop-erased random walks and uniform spanning trees on large finite graphs, revealing convergence to Brownian continuum random trees in high dimensions.

Contribution

It establishes the limiting distribution of loop-erased random walk lengths and proves the convergence of uniform spanning trees to Brownian CRTs on certain graphs.

Findings

Limiting distribution of loop-erased random walk lengths identified

Uniform spanning trees converge to Brownian continuum random trees in high dimensions

Results apply to graphs including the discrete torus in dimensions 5 and above

Abstract

Let x and y be chosen uniformly in a graph G. We find the limiting distribution of the length of a loop-erased random walk from x to y on a large class of graphs that include the discrete torus in dimensions 5 and above. Moreover, on this family of graphs we show that a suitably normalized finite-dimensional scaling limit of the uniform spanning tree is a Brownian continuum random tree.