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

• 成果类型: Article
• 署名作者: Schweinsberg, Jason
• 署名单位: University of California System; University of California San Diego
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-008-0149-7
• 发表日期: 2009
• 页码: 319-370
• 关键词: continuum random tree; dimensions

摘要

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 >= 5, in combination with results of Lawler concerning intersections of four-dimensional random walks.