Entropic repulsion for the free field:: Pathwise characterization in d ≱ 3

We study concentration properties of the lattice free field {phi(chi)}(chi epsilon Z)(d) in d greater than or equal to 3, i.e. the centered Gaussian field with covariance given by the Green function of the (discrete) Laplacian, when constrained to be positive in a region of volume O(N-d) (hard-wall condition). It has been shown in [3] that, as N --> infinity, the conditioned field is pushed to infinity: more precisely the typical value of the phi-variable to leading order is c root logN, and the exact value of c was found. It was moreover conjectured that the conditioned field, once this diverging height is subtracted, converges weakly to the lattice free field. Here we prove this conjecture, along with other explicit bounds, always in the direction of clarifying the intuitive idea that the free field with hard-wall conditioning merely translates away from the hard wall. We give also a proof, alternative to the one presented in [3], of the lower bound on the probability that the free field is everywhere positive in a region of volume N-d.