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 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.
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Supported in part by NSF Grant DMS-0504882.
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Schweinsberg, J. The loop-erased random walk and the uniform spanning tree on the four-dimensional discrete torus. Probab. Theory Relat. Fields 144, 319–370 (2009). https://doi.org/10.1007/s00440-008-0149-7
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DOI: https://doi.org/10.1007/s00440-008-0149-7