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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 $\eta_1 \le 1/3$ for continuous Gaussian fields with rapid
correlation decay (e.g. the Bargmann-Fock field), and in $d \ge 3$ we prove
$\eta_1 \le d/3$ for finite-range fields, both discrete and continuous, and
$\eta_1 \le d-2$ 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.

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