Consider the centered Gaussian field on Zd, d⩾2l+1, with covariance matrix given by (∑j=lKqj(−Δ)j)−1 where Δ is the discrete Laplacian on Zd, 1⩽l⩽K and qj∈R,l⩽j⩽K are constants satisfying ∑j=lKqjrj>0 for r∈(0,2] and a certain additional condition. We show the probability that all spins are positive in a box of volume Nd decays exponentially at a rate of order Nd−2l log N 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 Zd (i.e., K=l=1,q1=1).

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We consider a class of d-dimensional Gaussian lattice field which is known as a model of semi-flexible membrane. We study the free energy of the model with external potentials and show the following:(1) We consider the model with δ-pinning and prove that the field is always localized when d ≥ 4.(2) Consider the model confined between two hard walls. We give asymptotics of the free energy as the height of the wall goes to infinity.

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We consider a class of effective models on Z d called Gaussian membrane models with square-well pinning or δ-pinning. It is known that when d = 1 this model exhibits a localization/delocalization transition that depends on the strength of the pinning. In this paper, we show that when d ≥ 2, once we impose weak pinning potentials the field is always localized in the sense that the corresponding free energy is always positive. We also discuss the case that both square-well potentials and repulsive potentials are acting in high dimensions.