The scaling limit of the membrane model

TL;DR

This paper proves that the discrete membrane model converges to a continuum limit in dimensions two and higher, revealing different regularity properties and scaling behaviors depending on the dimension.

Contribution

It establishes the scaling limit of the discrete membrane model in dimensions two and above, extending previous results and characterizing the limit as a random field or distribution.

Findings

In $d=2,3$, the limit is a Hölder continuous random field.

In $d extgreater 3$, the limit is a random distribution.

The maximum scaling limit is derived for $d=2,3$.

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 $d\ge 2$. Namely, it is shown that the scaling limit in $d=2,\,3$ is a H\"older continuous random field, while in $d\ge 4$ 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 (2009) in $d=1$.