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

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Abstract

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).

Original languageEnglish
Pages (from-to)2939-2951
Number of pages13
JournalJournal of Mathematical Physics
Volume44
Issue number7
DOIs
Publication statusPublished - 2003 Jul 1

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ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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