A 0-1 law for the massive Gaussian free field

TL;DR

This paper establishes a sharp phase transition for the level sets of a massive Gaussian free field, showing polynomial decay of crossing probabilities below a certain threshold, despite long-range dependencies.

Contribution

It introduces a novel differential inequality and influence theorem to analyze phase transition sharpness in a correlated percolation model with long-range dependence.

Findings

Crossing probabilities converge to 1 polynomially fast below threshold h_{**}

Phase transition is sharp despite long-range dependence

Identifies critical thresholds h_{*} and h_{**} for percolation

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