Cluster capacity functionals and isomorphism theorems for Gaussian free fields

• 成果类型: Article
• 署名作者: Drewitz, Alexander; Prevost, Alexis; Rodriguez, Pierre-Francois
• 署名单位: University of Cologne; University of Cambridge; Imperial College London
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-021-01090-0
• 发表日期: 2022
• 页码: 255-313
• 关键词: random interlacements; percolation; set

摘要

We investigate level sets of the Gaussian free field on continuous transient metric graphs g and study the capacity of its level set clusters. We prove, without any further assumption on the base graph G, that the capacity of sign clusters on (G) over tilde is finite almost surely. This leads to a new and effective criterion to determine whether the sign clusters of the free field on (G) over tilde are bounded or not. It also elucidates why the critical parameter for percolation of level sets on (G) over tilde vanishes in most instances in the massless case and establishes the continuity of this phase transition in a wide range of cases, including all vertex-transitive graphs. When the sign clusters on (G) over tilde do not percolate, we further determine by means of isomorphism theory the exact law of the capacity of compact clusters at any height. Specifically, we derive this law from an extension of Sznitman's refinement of Lupu's recent isomorphism theorem relating the free field and random interlacements, proved along the way, and which holds under the sole assumption that sign clusters on (G) over tilde are bounded. Finally, we show that the law of the cluster capacity functionals obtained in this way actually characterizes the isomorphism theorem, i.e. the two are equivalent.