The scaling limit of the membrane model

Alessandra Cipriani, Biltu Dan, Rajat Subhra Hazra

Research output: Contribution to journalArticleScientificpeer-review

6 Citations (Scopus)

Abstract

On the integer lattice, we consider the discrete membrane model, a random interface in which the field has Laplacian interaction. We prove that, under appropriate rescaling, the discrete membrane model converges to the continuum membrane model in d ≥ 2. Namely, it is shown that the scaling limit in d = 2, 3 is a Holder continuous random field, while in d ≥ 4 the membrane model converges to a random distribution. As a by-product of the proof in d = 2, 3, we obtain the scaling limit of the maximum. This work complements the analogous results of Caravenna and Deuschel (Ann. Probab. 37 (2009) 903-945) in d = 1.

Original languageEnglish
Pages (from-to)3963-4001
Number of pages39
JournalAnnals of Probability
Volume47
Issue number6
DOIs
Publication statusPublished - 2019

Keywords

  • Continuum membrane model
  • Green's function
  • Membrane model
  • Random interface
  • Scaling limit

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