2021/02/24 by Vivek Dewan, Dewan, Vivek, Stephen Muirhead +1 · 1 citation
Mathematics · #2010 MSC classification: 60G60 (primary) #60F99 (secondary) #FOS: Mathematics #FOS: Physical sciences #Markov Chains and Monte Carlo Methods #Mathematical Physics (math-ph) #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics
paper · doi:10.48550/arxiv.2102.12123
openalex publication_date 2021/02/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove upper bounds on the one-arm exponent η1 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 η1 ≤ 1/3 for continuous Gaussian fields with rapid correlation decay (e.g. the Bargmann-Fock field), and in d ≥ 3 we prove η1 ≤ d/3 for finite-range fields, both discrete and continuous, and η1 ≤ 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.