A 0-1 law for the massive Gaussian free field

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 $h_{\ast }$ for percolation, a second parameter $h_{\ast \ast }\ge h_{\ast }$ characterizes a strongly subcritical regime. We prove that the relevant crossing probabilities converge to $1$ polynomially fast below $h_{\ast \ast }$, 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.

arXiv comment24 pages, 1 figure, to appear in Probab. Theor. Rel. Fields