ENTROPIC REPULSION OF THE LATTICE FREE-FIELD

Consider the massless free field on the d-dimensional lattice Z(d),d g reater 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 wa lk on Z(d). We show that the probability, that all the spins are posit ive 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 t ransient random walks with transition functions in the domain of the n ormal and alpha-stable law.