Properties of self-avoiding walks

The total number of SAWs of N steps, starting from point (f) is  

This series converges at lar r and diverges when r reaches the critical value 

Having located the transition point, we return to the general correlation (5 5j and rewrite it (for small c) in the form 

Thus the relationship between number of paths and magnetic correla- 

The limiting behavior of the correlations at large distances is given by an Omstein-Zemike form (introduced in Section X.1.4). Inverting the Laplace transformation [eq. (X.39)], it is then possible to fmd the asymptotic form of or of pp in eq. (X.40) at large r. The result is  

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Despite the shortage of rigorous analytical results, information has been accumulating regarding various properties of self-avoiding walks, and it now seems possible to put forward a coherent pattern of general behavior. This is the aim of the present paper. No attempt will be made to provide complete references to the extensive literature, but a sufficient number of key papers will be cited to enable anyone interested to do so for himself,... 

Four independent approaches have been used to investigate the properties of self-avoiding walks. 

In his paper Domb presents a detailed analysis of the statistical properties of self-avoiding walks on lattices.1 These walks serve as models for linear polymer chains with hard-core intramolecular interactions associated with the exclusion of multiple occupancies of the lattice sites by the chain so-called chains with excluded volume. 

In the case of flexible polymer chains, the dominant problem for theory has been the understanding of the statistical properties of self-avoiding random walks in 2, 3 and 4 dimensions. The case of D>3 arises of course because the self-avoiding interaction only becomes perturbative in 4 dimensions and... 

The actual properties of self-avoiding polymers have been studied by detailed calculations, by computer simulations, and by measurements on real polymer solutions. All these approaches confirm that these polymers, like random-walk polymers, are fractals. They have a D very close to 5/3, a value consistent with des Cloizeaux s inequality. 

A self-avoiding walk corresponds to the limit of a restricted walk as r- oo. Such a walk is no longer Markovian, and no general analytical methods have been found for determining its properties. However, evidence has been accumulating to indicate that the basic characteristics differ in an essential manner from those of restricted walks. We now proceed to discuss this evidence in further detail. 

We may now inquire how the fundamental change in properties takes place in passing from a restricted to a self-avoiding walk as r- co. For example, taking the simplest property, the total number of walks, how do we pass from the complex formula (10) for restricted walks to the simple formula (16) for self-avoiding walks, and what is the significance of the factor w9 in Eq. (16) ... 

To obtain a clearer insight into the relationship between the Ising and self-avoiding walk enumerations, it is better to pass to the logarithm of the partition function, In Z, since this eliminates the need to consider disjoint graphs. We can then find a property of the Ising model which corresponds to each of C , u , (R 2), f (x) dx. The Ising enumerations are much more... 

Since the critical behavior of PS models is essentially determined by the properties of loops, PS [70,71], and later Fisher [72], were led to consider various loop classes (together with straight paths for the double stranded segments). Whereas PS analyzed loop classes derived from random walks, Fisher considered loop classes derived from self-avoiding walks. 

The properties of a polymer chain with self-attraction on a fractal were first studied by Klein and Seitz [34]. They used the self-avoiding walks on the Sierpinski gasket, which is the 6 = 2 member of the Given-Mandelbrot family. We consider below the case of 3-simplex, which is somewhat simpler to treat. 

The properties of molt cular chain conformations or of random self-avoiding walks of linked segments belong to the class of universality with d = 3 and n = 0. It was first established by do Gennes, who brought the parameters of the magnetic problem into agreement with those of the polymer problem ... 

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