Iterative renormalization of polymers on a lattice

The iterative renormalization method on lattices was extended to the case of polymers by H. Hilhorst in 1976.8 The technique used by Hilhorst relies on the polymer-magnetic system correspondence for n - 0, as it is described in Chapter 11, Section 3.2. Hilhorst introduced spins located on the sites of a cyclic triangular lattice a spin rM with components (Tm corresponds to each lattice site M. The components take the values — n1/2, 0 or n1/2 and have to fulfil the condition that only one of these components is different from zero. The Hamiltonian of the system is given by the sum 

Renormalization consists in grouping the lattice sites three by three so as to form a new triangular lattice whose mesh is /Z times larger (see Fig. 12.1). Thus, let us consider three sites A, B, C bearing spins ta, tb, ac we can associate with them a site bearing a spin ctabc which is a random vector with the same properties as the rM. The probability law which Hilhorst attributes to rABC 

the value of the exponent v found in the limit n- 0 by Hilhorst is v = 0,740. However, this approach is not completely satisfactory. As happens for many of these iterative methods, the difficulty comes from the fact that the choice of new variables at each iteration is somewhat arbitrary. A rule must be adopted (as was done by Hilhorst) but no criterion exists for determining this rule in a unique way. Thus, the method lacks efficiency and reliability. 

It is also possible to ignore the polymer-magnetic system correspondence and to apply the renormalization process directly to the polymer-lattice system,9 but the results thus obtained have not been very stimulating. 

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