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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