ENTROPIC REPULSION OF THE LATTICE FREE-FIELD

Consider the massless free field on the d-dimensional lattice Z(d),d greater than or equal to 3; that is the centered Gaussian field on R(Zd) with covariances given by the Green function of the simple random walk on Z(d). We show that the probability, that all the spins are positive in a box of volume N-d, decays exponentially at a rate of order N-d-2 logN and compute explicitly the corresponding constant in terms of the capacity of the unit cube. The result is extended to a class of transient random walks with transition functions in the domain of the normal and alpha-stable law.