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

TL;DR

This paper proves that the interface model with mixed gradient and Laplacian interactions converges to the Gaussian free field in any dimension, using Fourier analysis and discrete PDE techniques.

Contribution

It establishes the scaling limit of the $( abla+ riangle)$-model as the Gaussian free field across all dimensions, combining Fourier and PDE methods.

Findings

Scaling limit is the Gaussian free field in any dimension.

Uses Fourier analysis for infinite volume cases.

Employs discrete PDE techniques for finite volume cases.

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.