A Renormalization Group Analysis of the Hierarchical Model by Pierre Collet

By Pierre Collet

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Poly~omial , and again a fixed point theorem applies. We do not expect this to happen at the points I C = 2 (2J+i)/2 , because t~ere the natural ansatz is e x p ( ~ a x 2j+l) which is unbounded for x ~ + ~ . For the case c near 2 ~/j all prece- ding results hold, so that one has the bifurcation picture space of Fig. 2 O 1 2 I/3 21/2 2 where each branch is controlled near the branching point. These branches correspond to "critical", "tricrltlcal", "tetrac~itical" ... behaviour, 34 as c = 2 I/2, 2 I/3 , 2 I/4 ...

Then we get log < SoS2J > 2N'+J f lim = J ~ ~ _ log2c log 2 j s i n c e ~ p ~) (f) converges to the image of ~c By definition log lim J ~ ~ Remarks = " 2 - d - ~ " , so that "~" = i + log2c log J on Section 5: Our treatment part, < SoS j > an expansion is a precise version of standard of the discussion in [18] arguments, and, in . 6. Global properties of the F l o w So far, we have regarded the action of the renormalization group as a purely local phenomenon in the space of densities ~. In this section we describe the mathematics of the action of the renormalization group in the large while the next section will be devoted to the physical implications of the global properties of the RG.

Discussion of the Critical Indices One of the triumphs in the RG approach has been the correct prediction of experimentally measured critical indices. The critical indices are defined as follows : Let Q(6) be some physical quantity de- pending on the inverse temperature 6 = I/kT, where k is the Boltzmann constant. e. (or diverge) as ~ ~c " Then the critical index of Q a t ~c (from above or below) is the limit (if it exists) VQ = lim ~ Note that in particular if ges as 6 ~ 6c log Q(~) / log IB-~c I ~c VQ ~ 0 , then this means that Q(~) diver- and ~Q measures in some sense how fast this diver- gence is.

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A Renormalization Group Analysis of the Hierarchical Model by Pierre Collet
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