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

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