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

Abstract

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).