Semi-dilute limit and field theory

We showed that, in the semi-dilute limit we must have 

Actually, a true justification of (13.2.83) could be obtained only through an elaborate study of the structure of the diagrams and of their contributions, in the semi-dilute limit. Such a study has not been made yet. In what follows, we shall 

In Chapter 11, we showed that for an equilibrium ensemble, the osmotic pressure /7, the polymer concentration C and the area concentration are connected by the following relations [see (11.5.43)] 

In this formula, T(M) is the generating function of the vertex functions (in the limit n = 0). It depends on parameters a and M or more precisely on (a — ac) and M because the short distance cut-off can be absorbed in the constant ac. Moreover, we note that the magnetic field B is related to T(M) by the equality [see (11.5.14)] 

In the limit a ae, the vertex functions diverge, but the divergences can be absorbed by renormalization. Thus, in agreement with (12.3.3), we set 

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