The loop-erased random walk and the uniform spanning tree on the four-dimensional discrete torus |
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Authors: | Jason Schweinsberg |
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Institution: | 1. Department of Mathematics, 0112 University of California at San Diego, 9500 Gilman Drive, La Jolla, CA, 92093-0112, USA
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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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