The scaling limit of the membrane model

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 $d=1$.