In this chapter we study self avoiding surfaces on a lattice. These surfaces are not immediately relevant for the study of polymers, although they could be of interest in the study of β-sheet polymers, which are important building blocks in proteins. Surfaces are used as models in the study of membranes or interfaces. Moreover, as we will see below, they allow the generalisation of vesicles to d = 3. The reason why we discuss surfaces in this book is mainly to show how the methods introduced in the statistical mechanics of polymers can be used in the study of other, but related, problems.

Several kinds of surfaces have been introduced in the literature. A distinction has to be made between surfaces which are models of polymerised membranes and those which describe liquid membranes. In the first case, the number of nearest neighbours of a given monomer is fixed. An interesting model is that of so called tethered surfaces introduced by Kantor, Kardar and Nelson. For liquid surfaces, on the other hand, the number of neighbours is not fixed. In this chapter, we will limit ourselves to a study of a lattice model of liquid surfaces, the ‘plaquette’ surfaces.

The critical behaviour of these surfaces is closely related to that of branched polymers. This is one of the many relations between surfaces and polymers.

Let us begin by defining the objects which we will study in this chapter and which will be referred to as plaquette surfaces or as self avoiding surfaces (SAS). An example of such a surface is shown in figure 11.1. The surface is built out of plaquettes of the cubic lattice.