By Pierre Collet
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5). the following conditions. 1. ci)-c5) z az % ( z ) / (%(z)) ~ ~ L There are o n ~ s , . functions ~o ~ ~ satisfying (Proof : Section 16). The choice of ~crit and ~o determines a Hamiltonian ~ = ~N, fo through the formula ! 2) Ir 46 For any such Hamiltonian we shall calculate the critical indices. They do not depend on the particular choice of ~o and this fact is called universality. Since all our discussions are in terms of the functions ~ , we shall describe now how the temperature dependence of the model is reflected in the space of the functions ~ .
8) 51 We see that the linearization FM = I___ F L 2N,~N,~(~N ,- ) I__ 2N 2N Therefore, 1,~N,J'TLN(&~N ) FL as is nothing else than an integral and it depends continuously on 0(I) in L log llm N ~ ~ 8~crit ( % ) l°gI~N - ~critl 1 log y FL 1,~N, aoe ° = lim N ~ ~ Anticipating = N lim ~ ~ , and is not zero. 20) index is I in this case. We shall now do the analogous lity and the magnetization tibility is defined as calculations for the susceptibi- with somewhat less details . 21) f N H dSj j=l exp( - 6 ~N, f) The normalized expectations necessitate a discussion of 9 , the nor- malized RG transformation introduced already in Eq.
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A Renormalization Group Analysis of the Hierarchical Model in Statistical Mechanics by Pierre Collet