Transfer current and pattern fields in spanning trees

• 成果类型: Article
• 署名作者: Kassel, Adrien; Wu, Wei
• 署名单位: Swiss Federal Institutes of Technology Domain; ETH Zurich; Brown University
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-014-0588-2
• 发表日期: 2015
• 页码: 89-121
• 关键词: abelian sandpile model; conformal-invariance; Scaling Limit

摘要

When a simply connected domain () is approximated in a good way by embedded connected weighted graphs, we prove that the transfer current matrix (defined on the edges of the graph viewed as an electrical network) converges, up to a local weight factor, to the differential of Green's function on . This observation implies that properly rescaled correlations of the spanning tree model and correlations of minimal subconfigurations in the abelian sandpile model have a universal and conformally covariant limit. We further show that, on a periodic approximation of the domain, all pattern fields of the spanning tree model, as well as the minimal-pattern (e.g. zero-height) fields of the sandpile, converge weakly in distribution to Gaussian white noise.