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

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 Gaussian fields, arguments apply to other in universality class, including Poisson–Voronoi and Poisson–Boolean percolation. More precisely, dimension $$d=2$$ we that _1 \le 1/3$$ continuous fields with rapid correlation decay (e.g. Bargmann–Fock field), $$d \ge 3$$ d/3$$ finite-range both discrete continuous, d-2$$ decay. Although these results are classical (indeed they best-known general), existing proofs do not extend models, develop new approach based exploration relative entropy arguments. The proof also makes use Russo-type inequality sharpness phase transition mean-field bound fields.