# A Renormalization Group Analysis of the Hierarchical Model by Pierre Collet

By Pierre Collet

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In other words, the map can be made to coincide with the tangent map in the unstable directions on the even subsp~ce. The above construction is explicit and can in principle be calculated in perturbation theory. In the l i t e r a t u r e on the RG, the tangent vectors to the unstable manifold away from ~ are called the relevant scaling fields and thus the above procedures allow to compute the hi~her order corrections -log ~ to the scaling fields as a function of ~ ~ L (or which would be called the "Hamiltonian").

Llm exists and is different h ~ 0 M6'h'f from zero. This will only be the case below the critical (6 > 6crit ), as we shall see later. 35) near the critical in the two phase region. Summarizing, near the critical we see that the behaviour of the various quantities temperature completely controlled or near the Critical by the llnearization field (h = O) is of the tangent m a p ~ j ~ ( ~ c ). 57 Since we have seen that ~c is C ~ analytically in e ~ 0, and since 9J~e(e~) depends on e~ , the (isolated) elgenvalues of ~J~c(~) have asymp- I totic expansions (actually, k ° = 2, k I = 2/c w , so that this statement Is only relevant for k2).

M. : Critical BLEHER indices for models with long range forces (Numerical Calculations). Preprint. Inst. , Acad. Sci. SSSR (1975). M. : A second order phase transition in some ferroma- BLEHER gnetic models. Trudy Mosc. Math. 0bshestvo33, 155 (1975). M. G. SINAI : Critical indices for systems with slowly decaying interaction. Eksp. Teor. Fiz. 67 391 (1974) [Sov. Phys. JETP. 40 , 195 (i975)]. 8. is a variant of an argument suggested by Nappi-Hegerfeldt and given in [23] M. CASSANDRO, G. JONA-LASINIO : Asymptotic behaviour of the auto covariance function and violation of strong mixing (Preprint).