Numerical data on self-avoiding walks

We see that the direct data on coils are not quite conclusive. It is then helpful to return to theoretical calculations. There do exist rather accurate numerical studies on real chains on a lattice. The chain is still represented by a random walk as in Fig. (1.1), but the main difference is that now this walk can never intersect itself. We call it a self-avoiding walk (SAW). 

The mathematical properties of simple random walks are trivial, but the mathematical properties of SAWs are complex. Two numerical methods have been used to study the SAWs  

All these studies have been performed on three-dimensional lattices and in other dimensionalities, d. The case for = 1 corresponds to chains along a line and is simple. The case ford = 2 may physically correspond to chains adsorbed at an interface. Higher dimensionalities (d = 4, 5...) are also of interest for the theorist, although they do not correspond to realizable systems. One important advance (during the past 10 years) has been to recognize the interest of discussing any statistical problem in arbitrary dimensions and to classify systems according to their behavior as a function of d. Thus, we shall often keep d as a parameter in our discussion of polymer chains. 

The results of numerical studies on SAWs are usefully summarized in a recent review by McKenzie. Our presentation, however, is slightly different since the physical meaning of the essential exponents has become more apparent in the recent years. 

The total number of SAWs ofN steps has the asymptotic form (at large )V) 

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