Abstract: We prove upper bound bounds on the one-arm exponent 
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 university class, including Poisson-Voronoi and Poisson-Boolean percolation. More precisely, in dimension 
we prove 
for Gaussian fields with rapid correlation decay (e.g. the Bargmann-Fock field) and in dimension 
we prove 
for finite-range fields and 
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.
In the first talk Vivek will present the main results and sketch the proof in the simpler setting of Bernoulli percolation. In the second talk Stephen will discuss how to adapt the arguments to the Gaussian setting, and present further applications to critical exponent inequalities.