First passage sets of the 2D continuum Gaussian free field

• 成果类型: Article
• 署名作者: Aru, Juhan; Lupu, Titus; Sepulveda, Avelio
• 署名单位: Swiss Federal Institutes of Technology Domain; Ecole Polytechnique Federale de Lausanne; Centre National de la Recherche Scientifique (CNRS); Sorbonne Universite; Universite Paris Cite; Centre National de la Recherche Scientifique (CNRS); Ecole Centrale de Lyon; Institut National des Sciences Appliquees de Lyon - INSA Lyon; Universite Claude Bernard Lyon 1; Universite Jean Monnet; CNRS - National Institute for Mathematical Sciences (INSMI)
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-019-00941-1
• 发表日期: 2020
• 页码: 1303-1355
• 关键词: multiplicative chaos; level lines; reversibility; sle

摘要

We introduce the first passage set (FPS) of constant level -a of the two-dimensional continuum Gaussian free field (GFF) on finitely connected domains. Informally, it is the set of points in the domain that can be connected to the boundary by a path on which the GFF does not go below -a. It is, thus, the two-dimensional analogue of the first hitting time of -a by a one-dimensional Brownian motion. We provide an axiomatic characterization of the FPS, a continuum construction using level lines, and study its properties: it is a fractal set of zero Lebesgue measure and Minkowski dimension 2 that is coupled with the GFF Phi as a local set A so that Phi+a restricted to A is a positive measure. One of the highlights of this paper is identifying this measure as a Minkowski content measure in the non-integer gauge r?|log(r)|1/2r2, by using Gaussian multiplicative chaos theory.