On decoupling inequalities and percolation of excursion sets of the Gaussian free field

TL;DR

This paper establishes decoupling inequalities for the Gaussian free field on integer lattices and demonstrates exponential decay of the connectivity function of excursion sets at high thresholds, with specific corrections in three dimensions.

Contribution

It introduces new decoupling inequalities for the Gaussian free field and applies them to analyze the percolation properties of excursion sets.

Findings

Decoupling inequalities are proven for the Gaussian free field in dimensions d≥3.

Exponential decay of the connectivity function is shown for large thresholds.

Logarithmic correction appears in the decay rate for d=3.

Abstract

We prove decoupling inequalities for the Gaussian free field on $\mathbb{Z}^d$, $d\geq 3$. As an application, we obtain exponential decay (with logarithmic correction for $d=3$) of the connectivity function of excursion sets for large values of the threshold.