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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 n 2(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.

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