Logarithmic Corrections to the Alexander–Orbach Conjecture for the Four-Dimensional Uniform Spanning Tree

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 $$n^2 (\log n)^{-1/3+o(1)}$$ n 2 ( log n ) - 1 / 3 + o ( 1 ) , that the typical intrinsic displacement of an n-step random walk is $$n^{1/3} (\log n)^{1/9-o(1)}$$ n 1 / 3 ( log n ) 1 / 9 - o ( 1 ) , and that the n-step return probability of the walk decays as $$n^{-2/3}(\log n)^{1/9-o(1)}$$ n - 2 / 3 ( log n ) 1 / 9 - o ( 1 ) .