Entropic repulsion for a Gaussian lattice field with certain finite range interaction

Consider the centered Gaussian field on ℤ^d, d≥2l+1, with covariance matrix given by (Σ_j=l^Kq_j( - Δ)^j)^-1 where Δ is the discrete Laplacian on ℤ^d, 1 ≤ l ≤ K and q_j ∈ ℝ,l ≤ j ≤ K are constants satisfying Σ_j=l^Kq_jr^j>0 for r ∈ (0,2] and a certain additional condition. We show the probability that all spins are positive in a box of volume N^d decays exponentially at a rate of order N^d-2l logN and under this hard-wall condition, the local sample mean of the field is repelled to a height of order √log N. This extends the previously known result for the case that the covariance is given by the Green function of simple random walk on ℤ^d (i.e., K= l = 1,q₁ = 1).