  We prove upper bounds on the one-arm exponent $\eta_1$ for dependent
percolation models; while our main interest is level set percolation of smooth
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 $\eta_1 \le 1/3$ for
Gaussian fields with rapid correlation decay (e.g.\ the Bargmann-Fock field),
and in general dimensions we prove $\eta_1 \le d/3$ for finite-range fields 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. We
also establish a new Russo-type inequality for smooth Gaussian fields which we
use to prove the sharpness of the phase transition for finite-range fields.
