Logarithmic corrections to scaling in the four-dimensional uniform spanning tree

TL;DR

This paper calculates precise logarithmic corrections to mean-field scaling for the uniform spanning tree in four dimensions, revealing detailed probabilistic behaviors and implications for related models like the Abelian sandpile.

Contribution

It provides the first detailed computation of logarithmic corrections to mean-field scaling in four-dimensional uniform spanning trees, including concentration estimates for loop-erased random walk capacity.

Findings

Probability of the past containing a path of length n is of order (log n)^{1/3} n^{-1}

Probability of the past containing at least n vertices is of order (log n)^{1/6} n^{-1/2}

Probability the past reaches the boundary is of order (log n)^{2/3+o(1)} n^{-2}

Abstract

We compute the precise logarithmic corrections to mean-field scaling for various quantities describing the uniform spanning tree of the four-dimensional hypercubic lattice $\mathbb{Z}^4$. We are particularly interested in the distribution of the past of the origin, that is, the finite piece of the tree that is separated from infinity by the origin. We prove that the probability that the past contains a path of length $n$ is of order $(\log n)^{1/3}n^{-1}$, that the probability that the past contains at least $n$ vertices is of order $(\log n)^{1/6} n^{-1/2}$, and that the probability that the past reaches the boundary of the box $[-n,n]^4$ is of order $(\log n)^{2/3+o(1)}n^{-2}$. An important part of our proof is to prove concentration estimates for the capacity of the four-dimensional loop-erased random walk which may be of independent interest. Our results imply that the Abelian sandpile model also exhibits non-trivial polylogarithmic corrections to mean-field scaling in four dimensions, although it remains open to compute the precise order of these corrections.