Open Access
November 2019 The scaling limit of the membrane model
Alessandra Cipriani, Biltu Dan, Rajat Subhra Hazra
Ann. Probab. 47(6): 3963-4001 (November 2019). DOI: 10.1214/19-AOP1351


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 $d\ge2$. Namely, it is shown that the scaling limit in $d=2,3$ is a Hölder continuous random field, while in $d\ge4$ 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 $d=1$.


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Alessandra Cipriani. Biltu Dan. Rajat Subhra Hazra. "The scaling limit of the membrane model." Ann. Probab. 47 (6) 3963 - 4001, November 2019.


Received: 1 February 2018; Revised: 1 September 2018; Published: November 2019
First available in Project Euclid: 2 December 2019

zbMATH: 07212175
MathSciNet: MR4038046
Digital Object Identifier: 10.1214/19-AOP1351

Primary: 31B30 , 60G15 , 60J45 , 82C20

Keywords: continuum membrane model , Green’s function , membrane model , random interface , Scaling limit

Rights: Copyright © 2019 Institute of Mathematical Statistics

Vol.47 • No. 6 • November 2019
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