The scaling limit of the $(\nabla +\Delta )$-model

Abstract

In this article we study the scaling limit of the interface model on ${\mathbb{Z}}^d$ where the Hamiltonian is given by a mixed gradient and Laplacian interaction. We show that in any dimension the scaling limit is given by the Gaussian free field. We discuss the appropriate spaces in which the convergence takes place. While in infinite volume the proof is based on Fourier analytic methods, in finite volume we rely on some discrete PDE techniques involving finite-difference approximation of elliptic boundary value problems.

Significantly revised version. Convergence for infinite volume now shown in Besov-Hölder spaces. Finite volume convergence extended to cover interactions by higher powers of Laplacian