Statistical mechanics of gradient models

Abstract

In this thesis, we consider gradient models on the d-dimensional discrete lattice. These models serve as effective models for interfaces and are also known as <em>continuous Ising models</em> . The height of the interface is modelled by a random field which is a real-valued map from a finite subset of the lattice. The energy of a configuration is given by a potential which only depends on finite differences of the random fields. We impose a tilt on the interface by considering the finite subset as a box with periodic boundary condition and the potential with shifted input. We are interested in the behaviour of the finite-volume gradient Gibbs measure as the box tends to the whole lattice, in dependence on the tilt and the temperature. <br /> For the potential being a small non-convex perturbation of the quadratic interaction and for small tilt and small temperature we prove scaling of the model to the Gaussian free field, strict convexity of the surface tension and algebraic decay of the covariance. The method of the proof is a rigorous implementation of the renormalisation group method.