$C^2$ regularity of the surface tension for the $\nabla \phi$ interface model

We consider the $\nabla \phi$ interface model with a uniformly convex interaction potential possessing H\"older 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 $C^{2,\beta}$ for some $\beta>0$. 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.