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

We prove upper bounds on the one-arm exponent η 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 η 1 ≤ 1 / 3 for continuous Gaussian fields with rapid correlation decay (e.g. the Bargmann–Fock field), and in d ≥ 3 we prove η 1 ≤ d / 3 for finite-range fields, both discrete and continuous, and η 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.