Exceptional points of two-dimensional random walks at multiples of the cover time

• 成果类型: Article
• 署名作者: Abe, Yoshihiro; Biskup, Marek
• 署名单位: Chiba University; University of California System; University of California Los Angeles
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-022-01113-4
• 发表日期: 2022
• 页码: 1-55
• 关键词: planar brownian-motion; random interlacements; percolation; FIELDS

摘要

We study exceptional sets of the local time of the continuous-time simple random walk in scaled-up (by N) versions D-N subset of Z(2) of bounded open domains D subset of R-2. Upon exit from D-N, the walk lands on a boundary vertex and then reenters D-N through a random boundary edge in the next step. In the parametrization by the local time at the boundary vertex we prove that, at times corresponding to a theta-multiple of the cover time of D-N, the sets of suitably defined lambda-thick (i.e., heavily visited) and X-thin (i.e., lightly visited) points are, as N -> infinity, distributed according to the Liouville Quantum Gravity Z(lambda)(D) with parameter lambda-times the critical value. For theta < 1, also the set of avoided vertices (a.k.a. late points) and the set where the local time is of order unity are distributed according to Z(root)(theta)(D). The local structure of the exceptional sets is described as well, and is that of a pinned Discrete Gaussian Free Field for the thick and thin points and that of random-interlacement occupation-time field for the avoided points. The results demonstrate universality of the Gaussian Free Field for these extremal problems.