On pinned fields, interlacements, and random walk on (Z/NZ)2

• 成果类型: Article
• 署名作者: Rodriguez, Pierre-Francois
• 署名单位: 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/s00440-018-0851-z
• 发表日期: 2019
• 页码: 1265-1299
• 关键词: vacant set; brownian-motion; percolation; torus; times

摘要

We define two families of Poissonian soups of bidirectional trajectories onZ2, which can be seen to adequately describe the local picture of the trace left by a random walk on the two-dimensional torus (Z/NZ)2, started from the uniform distribution, run up to a time of order (NlogN)2 and forced to avoid a fixed point. The local limit of the latter was recently established inComets et al. (Commun Math Phys 343:129-164, 2016). Our construction proceeds by considering, somewhat in the spirit of statistical mechanics, a sequence of finite volume approximations, consisting of random walks avoiding the origin and killed at spatial scale N, either using Dirichlet boundary conditions, or by means of a suitably adjusted mass. By tuning the intensity u of such walks with N, the occupation field can be seen to have a nontrivial limit, corresponding to that of the actual random walk. Our construction thus yields a two-dimensional analogue of the random interlacements model introduced inSznitman (Ann Math 171(3):2039-2087, 2010) in the transient case. It also links it to the pinned free field in Z2, by means of a (pinned) Ray-Knight type isomorphism theorem.