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

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

2006-02-23

arxiv.org/abs/math/0602515v3

Mathematics

Probability

A few typos and minor errors corrected, some proofs simplified

Scientific paper

Let x and y be points chosen uniformly at random from $\Z_n^4$, 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 $\Z_n^4$ 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 $d \geq 5$, in combination with results of Lawler concerning intersections of four-dimensional random walks.

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