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 Z(d) or R-d, d >= 2, with correlations decaying at least algebraically with exponent alpha > 0, including the discrete Gaussian free field (d >= 3, alpha = d - 2), the discrete Gaussian membrane model (d >= 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 e (0, d]) in dimension d >= 3 except the Gaussian free field, for which sharpness was proven in a recent breakthrough 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.