A 0-1 law for the massive Gaussian free field

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 $h_{**} \geq h_{*}$ 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.