SUBSEQUENTIAL SCALING LIMITS OF SIMPLE RANDOM WALK ON THE TWO-DIMENSIONAL UNIFORM SPANNING TREE

• 成果类型: Article
• 署名作者: Barlow, M. T.; Croydon, D. A.; Kumagai, T.
• 署名单位: University of British Columbia; University of Warwick; Kyoto University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/15-AOP1030
• 发表日期: 2017
• 页码: 4-55
• 关键词: erased random-walks; brownian-motion; galton-watson; CONVERGENCE; GROWTH; cluster

摘要

The first main result of this paper is that the law of the (rescaled) two-dimensional uniform spanning tree is tight in a space whose elements are measured, rooted real trees continuously embedded into Euclidean space. Various properties of the intrinsic metrics, measures and embeddings of the subsequential limits in this space are obtained, with it being proved in particular that the Hausdorff dimension of any limit in its intrinsic metric is almost surely equal to 8/5. In addition, the tightness result is applied to deduce that the annealed law of the simple random walk on the two-dimensional uniform spanning tree is tight under a suitable rescaling. For the limiting processes, which are diffusions on random real trees embedded into Euclidean space, detailed transition density estimates are derived.