Consider the centered Gaussian field on ℤd, d≥2l+1, with covariance matrix given by (Σj=lKqj( - Δ)j)-1 where Δ is the discrete Laplacian on ℤd, 1 ≤ l ≤ K and qj ∈ ℝ,l ≤ j ≤ K are constants satisfying Σj=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 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 ℤd (i.e., K= l = 1,q1 = 1).
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Mathematical Physics