LOOP-ERASED RANDOM WALK AND POISSON KERNEL ON PLANAR GRAPHS

• 成果类型: Article
• 署名作者: Yadin, Ariel; Yehudayoff, Amir
• 署名单位: University of Cambridge; Technion Israel Institute of Technology
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/10-AOP579
• 发表日期: 2011
• 页码: 1243-1285
• 关键词: uniform spanning-trees; Invariance

摘要

Lawler, Schramm and Werner showed that the scaling limit of the loop-erased random walk on Z(2) is SLE(2). We consider scaling limits of the loop-erasure of random walks on other planar graphs (graphs embedded into C so that edges do not cross one another). We show that if the scaling limit of the random walk is planar Brownian motion, then the scaling limit of its loop-erasure is SLE(2). Our main contribution is showing that for such graphs, the discrete Poisson kernel can be approximated by the continuous one. One example is the infinite component of super-critical percolation on Z(2). Berger and Biskup showed that the scaling limit of the random walk on this graph is planar Brownian motion. Our results imply that the scaling limit of the loop-erased random walk on the super-critical percolation cluster is SLE(2).