### Abstract

Consider the centered Gaussian field on ℤ^{d}, d≥2l+1, with covariance matrix given by (Σ_{j=l}^{K}q_{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}^{K}q_{j}r^{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,q_{1} = 1).

Original language | English |
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Pages (from-to) | 2939-2951 |

Number of pages | 13 |

Journal | Journal of Mathematical Physics |

Volume | 44 |

Issue number | 7 |

DOIs | |

Publication status | Published - 2003 Jul 1 |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics