Upper bounds on the one-arm exponent for dependent percolation models

• 成果类型: Article
• 署名作者: Dewan, Vivek; Muirhead, Stephen
• 署名单位: Communaute Universite Grenoble Alpes; Universite Grenoble Alpes (UGA); University of Melbourne
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-022-01176-3
• 发表日期: 2023
• 页码: 41-88
• 关键词: phase-transition; critical-behavior; probabilities; inequalities; Boundedness; sharpness; cluster; trees; decay

摘要

We prove upper bounds on the one-arm exponent eta(1) for a class of dependent percolation models which generalise Bernoulli percolation; while our main interest is level set percolation of Gaussian fields, the arguments apply to other models in the Bernoulli percolation universality class, including Poisson-Voronoi and Poisson-Boolean percolation. More precisely, in dimension d = 2 we prove that eta(1) <= 1/3 for continuous Gaussian fields with rapid correlation decay (e.g. the Bargmann-Fock field), and in d >= 3 we prove eta(1) <= d/3 for finite-range fields, both discrete and continuous, and eta(1 )< d - 2 for fields with rapid correlation decay. Although these results are classical for Bernoulli percolation (indeed they are best-known in general), existing proofs do not extend to dependent percolation models, and we develop a new approach based on exploration and relative entropy arguments. The proof also makes use of a new Russo-type inequality for Gaussian fields, which we apply to prove the sharpness of the phase transition and the mean-field bound for finite-range fields.