We consider the ∇ϕ interface model with a uniformly convex interaction potential possessing Hölder continuous second derivatives. Combining ideas of Naddaf and Spencer with methods from quantitative homogenization, we show that the surface tension (or free energy) associated to the model is at least C^2,β for some β > 0. We also prove a fluctuation-dissipation relation by identifying its Hessian with the covariance matrix characterizing the scaling limit of the model. Finally, we obtain a quantitative rate of convergence for the Hessian of the finite-volume surface tension to that of its infinite-volume limit.

中文翻译：

---

∇ϕ 界面模型的表面张力的 C2 规律

我们考虑具有均匀凸相互作用势的 ∇ ϕ接口模型，该模型具有 Hölder 连续二阶导数。将 Naddaf 和 Spencer 的思想与定量均质化方法相结合，我们表明，对于某些β > 0 ，与模型相关的表面张力（或自由能）至少为C ^2,β。我们还通过识别来证明波动-耗散关系它的 Hessian 具有表征模型缩放限制的协方差矩阵。最后，我们获得了有限体积表面张力的 Hessian 到其无限体积极限的收敛速度。