Analytic renormalizations

This makes it possible to calculate the critical value xc and the exponent v as was explained above, and this leads to the result mentioned at the beginning of this section. 

More recently, strip methods have also been used by H. Saleur and D. Derrida (unpublished) to calculate y/v. They found results compatible with the value y/v = 43/24 ( = 1.79167) predicted by Nienhuis (see Section 4). 

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However, very interesting results were obtained by applying analytic renormalization methods to perturbation series. These techniques initiated by Wilson himself, were developed by many physicists, and, among them, Brezin, Le Guillou, and Zinn-Justin played an important role. 

Consequently, to describe a critical system, one has to use renormalization techniques1,2 which come into play quite naturally however, the renormalization concept itself can be considered from quite different points of view. Roughly, we can distinguish two kinds of approach iterative renormalizations and analytic renormalizations. 

General principles common to all analytical renormalization techniques... 

In brief, the analytic renormalization techniques apply to continuous models depending on very few bare parameters, and they aim at expressing measurable quantities directly in terms of macroscopic and observable fundamental parameters. Thus, in spite of a certain mathematical complexity, this approach appears as essentially realistic. 

Analytic renormalization methods were applied to the Landau-Ginzburg method (around 1970) but, originally for a space dimension close to 4, dimension 4 being marginal. However, it appeared later that analytical methods could also be used directly in dimension 3, as G. Parisi suggested in 1973 (Cargese Summer School). 

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