Article Type

Research Article

Publication Title

Annals of Probability

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.

First Page

3963

Last Page

4001

DOI

10.1214/19-AOP1351

Publication Date

1-1-2019

Comments

Open Access, Green

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