Logarithmic corrections to the Alexander-Orbach conjecture for the four-dimensional uniform spanning tree

TL;DR

This paper precisely characterizes the logarithmic corrections to the Alexander-Orbach conjecture for the four-dimensional uniform spanning tree, revealing detailed asymptotic behaviors of volume, random walk displacement, and return probability.

Contribution

It provides the first rigorous derivation of logarithmic corrections to the Alexander-Orbach conjecture in four dimensions for the uniform spanning tree.

Findings

Volume of intrinsic n-ball is approximately n^2 / (log n)^{1/3}

Typical displacement of n-step random walk is about n^{1/3} (log n)^{1/9}

Return probability decays as n^{-2/3} (log n)^{1/9}

Abstract

We compute the precise logarithmic corrections to Alexander-Orbach behaviour for various quantities describing the geometric and spectral properties of the four-dimensional uniform spanning tree. In particular, we prove that the volume of an intrinsic $n$-ball in the tree is $n^2 (\log n)^{-1/3+o(1)}$, that the typical intrinsic displacement of an $n$-step random walk is $n^{1/3} (\log n)^{1/9-o(1)}$, and that the $n$-step return probability of the walk decays as $n^{-2/3}(\log n)^{1/9-o(1)}$.