Random walk loop soups and conformal loop ensembles

• 成果类型: Article
• 署名作者: van de Brug, Tim; Camia, Federico; Lis, Marcin
• 署名单位: Vrije Universiteit Amsterdam; New York University; New York University Abu Dhabi; Chalmers University of Technology; University of Gothenburg
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-015-0666-0
• 发表日期: 2016
• 页码: 553-584
• 关键词: critical percolation; Invariance; exponents; plane

摘要

The random walk loop soup is a Poissonian ensemble of lattice loops; it has been extensively studied because of its connections to the discrete Gaussian free field, but was originally introduced by Lawler and Trujillo Ferreras as a discrete version of the Brownian loop soup of Lawler and Werner, a conformally invariant Poissonian ensemble of planar loops with deep connections to conformal loop ensembles (CLEs) and the Schramm-Loewner evolution (SLE). Lawler and Trujillo Ferreras showed that, roughly speaking, in the continuum scaling limit, large lattice loops from the random walk loop soup converge to large loops from the Brownian loop soup. Their results, however, do not extend to clusters of loops, which are interesting because the connection between Brownian loop soup and CLE goes via cluster boundaries. In this paper, we study the scaling limit of clusters of large lattice loops, showing that they converge to Brownian loop soup clusters. In particular, our results imply that the collection of outer boundaries of outermost clusters composed of large lattice loops converges to CLE.