WIAS Preprint No. 2301, (2016)

Exponential decay of covariances for the supercritical membrane model


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

2010 Mathematics Subject Classification

  • 60K35 31B30 39A12 60K37 82B41


  • membrane model, pinning, bilaplacian, decay of covariances




We consider the membrane model, that is the centered Gaussian field on Zd whose covariance matrix is given by the inverse of the discrete Bilaplacian. We impose a δ-pinning condition, giving a reward of strength ε for the field to be 0 at any site of the lattice. In this paper we prove that in dimensions d≥5 covariances of the pinned field decay at least exponentially, as opposed to the field without pinning, where the decay is polynomial. The proof is based on estimates for certain discrete weighted norms, a percolation argument and on a Bernoulli domination result.

Appeared in

  • Comm. Math. Phys., 353:3 (2017) pp. 1217--1240.

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