In this article we study the scaling limit of the interface model on \({{\,\mathrm{{\mathbb {Z}}}\,}}^d\) 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.

中文翻译：

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$$（\ nabla + \ Delta）$$（∇+Δ）-模型的比例极限

在本文中，我们研究了\（{{\，\ mathrm {{\ mathbb {Z}}} \，}} ^ d \）上的界面模型的比例极限，其中，哈密顿量由混合梯度和Laplacian相互作用给出。我们表明，在任何维度上，缩放限制都是由高斯自由域给出的。我们讨论发生收敛的适当空间。在无穷大的体积中，证明是基于傅立叶分析方法的；而在无穷大的体积中，我们依靠一些离散的PDE技术，这些技术涉及椭圆边界值问题的有限差分近似。