Entropic repulsion for a class of Gaussian interface models in high dimensions

Consider the centred Gaussian field on the lattice ZdZd, dd large enough, with covariances given by the inverse of ∑j=kKqj(−Δ)j, where ΔΔ is the discrete Laplacian and qj∈R,k≤j≤Kqj∈R,k≤j≤K, the qjqj satisfying certain additional conditions. We extend a previously known result to show that the probability that all spins are nonnegative on a box of side-length NN has an exponential decay at a rate of order Nd−2klogNNd−2klogN. The constant is given in terms of a higher-order capacity of the unit cube, analogously to the known case of the lattice free field. This result then allows us to show that, if we condition the field to stay positive in the NN-box, the local sample mean of the field is pushed to a height of order logN.