Abstract
We investigate the phase transition in a non-planar correlated percolation model with long-range dependence, obtained by considering level sets of a Gaussian free field with mass above a given height h. The dependence present in the model is a notorious impediment when trying to analyze the behavior near criticality. Alongside the critical threshold $$h_*$$ for percolation, a second parameter $$h_{**} \ge h_*$$ characterizes a strongly subcritical regime. We prove that the relevant crossing probabilities converge to 1 polynomially fast below $$h_{**}$$ , which (firmly) suggests that the phase transition is sharp. A key tool is the derivation of a suitable differential inequality for the free field that enables the use of a (conditional) influence theorem.
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