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 $d\ge2$. 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 $d=1$.
Citation
Alessandra Cipriani. Biltu Dan. Rajat Subhra Hazra. "The scaling limit of the membrane model." Ann. Probab. 47 (6) 3963 - 4001, November 2019. https://doi.org/10.1214/19-AOP1351
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