WIAS Preprint No. 2220, (2016)

Fast decay of covariances under delta-pinning in the critical and supercritical membrane model


  • Bolthausen, Erwin
  • Cipriani, Alessandra
  • Kurt, Noemi

2010 Mathematics Subject Classification

  • 31B30 39A12 60K35 60K37 82B41


  • membrane model, pinning, bilaplacian, decay of covariances




We consider the membrane model, that is the centered Gaussian field on Z^d whose covariance matrix is given by the inverse of the discrete Bilaplacian. We impose a delta-pinning condition, giving a reward of strength epsilon for the field to be 0 at any site of the lattice. In this paper we prove that in dimensions larger than 4 covariances of the pinned field decay at least stretched-exponentially, as opposed to the field without pinning, where the decay is polynomial in dimensions larger than 5 and logarithmic in 4 dimensions. The proof is based on estimates for certain discrete Sobolev norms, and on a Bernoulli domination result.

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