In the previous chapter we learned the importance of the O(n)-model for the study of polymers. In this chapter we will see how in two dimensions the critical behaviour of the O(n)-model has been determined exactly. The critical exponents of this model were first conjectured from renormalisation group arguments by Cardy and Hamber. These conjectures were then confirmed by Nienhuis using an approximate mapping onto the ‘Coulomb gas’. Since the Coulomb gas can be renormalised exactly, this lent support to the belief that the exponents of Cardy and Hamber were indeed exact. In 1986, Baxter succeeded in solving the O(n)-model exactly on the hexagonal lattice. His results, which were obtained in the thermodynamic limit, were later extended to finite systems by Batchelor and Blöte. In more recent years an O(n)-model on the square lattice has received a lot of attention since it has a very rich critical behaviour.

We conclude this chapter with a discussion of the SAW on fractal lattices.

The Coulomb gas approach to the SAW in d = 2

In this section we discuss how the O(n)-model (2.17) can be related to the Coulomb gas. This relation holds for the O(n)-model on the hexagonal lattice. It will give exact results for exponents ν and γ which because of universality should also hold on other lattices. Furthermore an exact value for μ follows from this mapping. When we talk about an exact solution we must note that the result is derived by nonrigorous means but that nevertheless it is generally believed that the result is the exact one. All numerical calculations furthermore support these conjectured values.