For dimers and other models of random surfaces, limit shapes appear when boundary conditions force a certain response of the system. The main mathematical tool to study these responses is a variational principle which states that the limiting profile of the system must maximizes the integral of an entropy function often named surface tension. As a consequence, the strict convexity of the surface tension plays a crucial role as it forces the asymptotic profile which maximizes this integral to be unique. In this talk we will show that all models of Lipschitz random surfaces which are stochastically monotonic must have a strictly convex surface tension (joint with Piet Lammers).