Behaviour of P r for large values

In this section, we show that the function f(x) which appears in the expression (10.1.66) of P(r) has the following asymptotic behaviour for large values of x 

Here 5 and k are critical exponents which depend only on the dimension d of the space in which the polymer chain is embedded. We already indicated, in Chapter 3, Section 3.3, how, in 1966, Fisher14 found the relation 

In this section, we study this question again by starting from different premises. 

let us consider field theory. If the Lagrangian theory defined in Chapter 11 is a genuine field theory, as we believe, the Green s function has a pole for k2 = - m2 (where m can be considered as the mass of the particle which the field describes) and also a cut for more negative values of k1. Thus, for large values of r, we must attribute a dominant role to this pole. At the pole, O(x) vanishes, and in the vicinity of this point we may set 

As we are interested only in large values of r/ , we can evaluate the integral directly by using the steepest descent method. At the saddle point, r = r/2 . We thus obtain 

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