Chains in one-dimensional space

A one-dimensional lattice is made of regularly-spaced points on a straight line. Drawing a self-avoiding chain by starting from an origin O is a trivial operation (see Fig. 3.15). For any value N of the number of links, it is possible to draw two chains of length 

The passage to the continuous limit is also trivial and gives 

we would like to know the properties of a self-intersecting chain with repulsive interaction. The universality principle leads us to postulate that, for such a chain (JV - oo), the following properties are valid  

The length of the chain being defined, as always, by the equality L = Nl, the asymptotic law giving the end-to-end probability distribution can be written in the form 

Though these properties have not been very rigorously proved, intuitive arguments show that they are satisfactory. Moreover, Balian and Toulouse12 showed that in the continuous case the properties of a chain can be studied precisely by starting from a Lagrangian representation of the problem (see Chapter 11) and by using a transfer matrix method. In particular, these authors verified that the critical exponents are v = 1 and y = 1 [see also the article by Thouless (1975)]. 3 

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