In d = 3 neither conformal invariance nor the Coulomb gas technique is very helpful in determining the critical behaviour of self avoiding walks. The transfer matrix can only reach up to small widths W, series can only get up to rather small N, and so on. So we have to look for different methods. There are essentially three of these. The first is probably the most obvious one; we can perform experiments on real polymers. In this book, we will only mention the results of these. Secondly, a powerful numerical method which so far has not been discussed is the Monte Carlo technique. It can of course be applied more easily and more accurately in d = 2, but in d = 3 it hats less competition from other methods. That's why we will discuss it in section 4.2 mainly from the point of view of learning about polymers in d = 3. We begin the chapter with a brief discussion of the third method, which is the RG approach to the critical behaviour of polymers.

Direct renormalisation of the Edwards model

We already encountered the exact RG methods for SAWs on fractal lattices in the previous chapter. But such a real space approach can only work very approximately on Euclidean lattices. The most precise RG calculations for polymers are performed with continuum techniques. A first method uses the O(n)-model, calculates the exponents of that model using techniques such as the ∈-expansion, and then sends n → 0 in the final equations. Here we will not get into these calculations; they have been very well described elsewhere.