Percolation of strongly correlated Gaussian fields II. Sharpness of the phase transition

Abstract

We establish the sharpness of the phase transition for a wide class of Gaussian percolation models, on ${\mathbb{Z}}^{\mathit{d}}$ or ${\mathbb{R}}^{\mathit{d}}$, $\mathit{d}\ge 2$, with correlations decaying at least algebraically with exponent $\mathit{\alpha }>0$, including the discrete Gaussian free field ($\mathit{d}\ge 3$, $\mathit{\alpha }=\mathit{d}-2$), the discrete Gaussian membrane model ($\mathit{d}\ge 5$, $\mathit{\alpha }=\mathit{d}-4$), and many other examples both discrete and continuous. In particular, we do not assume positive correlations. This result is new for all strongly correlated models (i.e., $\mathit{\alpha }\in (0,\mathit{d}]$) in dimension $\mathit{d}\ge 3$ except the Gaussian free field, for which sharpness was proven in a recent breakthrough by Duminil-Copin et al. (Duke Math. J. 172 (2023) 839–913); even then, our proof is simpler and yields new near-critical information on the percolation density.

For planar fields which are continuous and positively correlated, we establish sharper bounds on the percolation density by exploiting a new ‘weak mixing’ property for strongly correlated Gaussian fields. As a byproduct, we establish the box-crossing property for the nodal set, of independent interest.

This is the second in a series of two papers studying level-set percolation of strongly correlated Gaussian fields, which can be read independently.