Abstract: We study the Gaussian free field on the Euclidean lattice in three and more dimensions and consider the percolation problem associated to its level-sets, as first investigated by Lebowitz and Saleur in 1986. We prove the equality of several natural critical parameters associated to this model. Our findings yield the sharpness of the associated phase transition, which corresponds to classical results in the context of Bernoulli percolation due to Menshikov and Aizenman-Barsky (in the subcritical phase) and Grimmett-Marstrand (in the supercritical phase).
Based on joint work with H. Duminil-Copin and S. Goswami.