Three-dimensional renormalization and calculation of critical exponents

2 Three-dimensional renormalization and calculation of critical exponents 

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). 

Calculations of the critical exponents v and y (or more precisely y and H = 2 — y/v) have been performed step by step. Around 1975, Nickel succeeded in obtaining,19 the coefficients of perturbation series up to order six or seven for several vertex functions. 

The diagrams were counted and their contributions were calculated with the help of a computer. Results were obtained for n = 0,1, 2, 3, 4. Incidentally we note that the study of the dependence in rt of the terms of the series is not 

For n = 0, which corresponds to the case of polymers (see Chapter 1IX the values obtained by Le Guillou and Zinn-Justin21 for the critical exponents, are (for d = 3) 

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