The Scaling Limit of the $$(\nabla +\Delta )$$-Model

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