Scaling limit of the loop-erased random walk Green's function

• 成果类型: Article
• 署名作者: Benes, Christian; Lawler, Gregory F.; Viklund, Fredrik
• 署名单位: City University of New York (CUNY) System; Brooklyn College (CUNY); University of Chicago; Royal Institute of Technology; Uppsala University
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-015-0655-3
• 发表日期: 2016
• 页码: 271-319
• 关键词: spanning-trees; partial sums; ising-model; approximation

摘要

We consider loop-erased random walk (LERW) running between two boundary points of a square grid approximation of a planar simply connected domain. The LERW Green's function is the probability that the LERW passes through a given edge in the domain. We prove that this probability, multiplied by the inverse mesh size to the power 3/4, converges in the lattice size scaling limit to (a constant times) an explicit conformally covariant quantity which coincides with the Green's function. The proof does not use SLE techniques and is based on a combinatorial identity which reduces the problem to obtaining sharp asymptotics for two quantities: the loop measure of random walk loops of odd winding number about a branch point near the marked edge and a spinor observable for random walk started from one of the vertices of the marked edge.