The scaling limit of the membrane model

Authors

Alessandra Cipriani, Delft University of Technology
Biltu Dan, Indian Statistical Institute, Kolkata
Rajat Subhra Hazra, Indian Statistical Institute, Kolkata

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

Recommended Citation

Cipriani, Alessandra; Dan, Biltu; and Hazra, Rajat Subhra, "The scaling limit of the membrane model" (2019). Journal Articles. 1008.
https://digitalcommons.isical.ac.in/journal-articles/1008