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

We establish the sharpness of the phase transition for a wide class of Gaussian percolation models, on $\mathbb{Z}^d$ or $\mathbb{R}^d$, $d \ge 2$, with correlations decaying at least algebraically with exponent $\alpha > 0$, including the discrete Gaussian free field ($d \ge 3, \alpha = d-2$), the discrete Gaussian membrane model ($d \ge 5, \alpha = d - 4$), 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. $\alpha \in (0,d]$) in dimension $d \ge 3$ except the Gaussian free field, for which sharpness was proven in a recent breakthrough by Duminil-Copin, Goswami, Rodriguez and Severo; 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.