Decoupling inequalities for the Ginzburg-Landau $\nabla \varphi$ models

We consider a class of of massless gradient Gibbs measures, in dimension greater or equal to three, and prove a decoupling inequality for these fields. As a result, we obtain detailed information about their geometry, and the percolative and non-percolative phases of their level sets, thus generalizing results obtained in arXiv:1202.5172, to the non-Gaussian case. Inequalities of similar flavor have also been successfully used in the study of random interlacements, see arXiv:1010.1490, arXiv:1212.1605. A crucial aspect is the development of a suitable sprinkling technique, which relies on a particular representation of the correlations in terms of a random walk in a dynamic random environment, due to Helffer and Sj\"{o}strand. The sprinkling can be effectively implemented by studying the Dirichlet problem for the corresponding Poisson equation, and quantifiying in how far a change in boundary condition along a sufficiently "small" part of the boundary affects the solution. Our results allow for uniformly convex potentials, and extend to non-convex perturbations thereof.