The loop-erased random walk and the uniform spanning tree on the four-dimensional discrete torus

Abstract

Let x and y be points chosen uniformly at random from \({\mathbb {Z}_n^4}\), the four-dimensional discrete torus with side length n. We show that the length of the loop-erased random walk from x to y is of order n2(log n)1/6, resolving a conjecture of Benjamini and Kozma. We also show that the scaling limit of the uniform spanning tree on \({\mathbb {Z}_n^4}\) is the Brownian continuum random tree of Aldous. Our proofs use the techniques developed by Peres and Revelle, who studied the scaling limits of the uniform spanning tree on a large class of finite graphs that includes the d-dimensional discrete torus for d ≥ 5, in combination with results of Lawler concerning intersections of four-dimensional random walks.