By Pierre Collet
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Extra resources for A Renormalization Group Analysis of the Hierarchical Model in Statistical Mechanics
In this section we show how the critical indices are obtained from the results of Section 4 . We define now the class of models for which the critical indices can be computed. These models are characterized by the fact that the one-spin part f in the Hamiltonian 3£N, f (Eq. 1) is a function which is near to - ~-i log 8~(~c) , the transform of ~c into the temperature dependent formulation (cf. 3). For other models, we cannot discuss the critical indices, because their mean-spin distributions might not fall into the neighborhood of we have "perfect" control ~ ~ , which is the only region in which .
J~ is bounded, 92~ Put in another way, neighborhoods does not are too big in function spaces. One could also say that the polynomial approximation @(~)~ I + a 04 does not push us sufficiently into the direction of the bifurcating branch for the implicit function thm. to apply. We shall therefore try a better initial approximation and this will be sufficient after some hard labour. 19) = fN(~,x) + remainder , where PN(C,x) is that polynomial in a,x such that fN(c,x) coincides with ~s Up to order N in s.
4 16 11 10 39 Note that k 4 is 1 - ~ leg 2 + 0 ( ~ 2) for S)J~(~ ) _ ponding while the corres- eigenvalue for ~)J~(i) is 2/c 2 ~ i + ~ l o g 2 . Therefore the eigenvalue for S)Tis smaller than one (~)Tis a contraction in the "direction" associated to k4) whil@ ~)~(1) is an expansion in the analogous direction. S o the bifurcating branch is in a sense more stable than the branch from which it bifurcates since the former has one more contractive direction than the latter. ). The "flow" T can be stretched by an infinitely differentiable coordinate transformation S so that the following is true.
A Renormalization Group Analysis of the Hierarchical Model in Statistical Mechanics by Pierre Collet