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Consider the massless free field on the d-dimensional lattice ℤ d , d≥3; that is the centered Gaussian field on ℝ ℤ d 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 α-stable law

Entropic repulsion of the lattice free field