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The loop-erased random walk and the uniform spanning tree on the four-dimensional discrete torus
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  • Published: 27 March 2008

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

  • Jason Schweinsberg1 

Probability Theory and Related Fields volume 144, pages 319–370 (2009)Cite this article

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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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Authors and Affiliations

  1. Department of Mathematics, 0112 University of California at San Diego, 9500 Gilman Drive, La Jolla, CA, 92093-0112, USA

    Jason Schweinsberg

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  1. Jason Schweinsberg
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Correspondence to Jason Schweinsberg.

Additional information

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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  • Received: 20 October 2006

  • Revised: 10 March 2008

  • Published: 27 March 2008

  • Issue Date: July 2009

  • DOI: https://doi.org/10.1007/s00440-008-0149-7

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Keywords

  • Loop-erased random walk
  • Uniform spanning tree
  • Continuum random tree

Mathematics Subject Classification (2000)

  • Primary: 60K35
  • Secondary: 60G50
  • 60D05
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