A Renormalization Group Analysis of the Hierarchical Model by Pierre Collet

By Pierre Collet

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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). C. HEGERFELDT : Prime field decompositions and infinitely divisible states on Borcher's tensor algebra. Commun. math. Phys. 4 5 , 137 (1975). 4. The Flew Around the 9Tlxed Point In this section , and the following, we fix ~ to some (sufficien- tly small) positive value.

6 is now basically as follows. Let 5 be the spectral projection corresponding to one of the elgenvalues ~fN, g By perturbation theory we have Ikj - 11-1 4 kj of Q(C1). Then fl (~fN, - 1) -1 ~gll= (kj- 1) -1 Pig I1~o (xj 1) -1 xj -n(~) I lf~fN,~n(e) PJgll°° (kj 1)-lkj -n(e) (Xj 1 )_lkj-n(e ) n(e)-i I o(1) II ~N,~ I 0(1) 4 n(e)-I II ~g I12~-z + 1,7 Pig I]2~/c-lc-n(~)+1+1,7 by a repeated application of (ii) . ,; -< o(~ -k) II g 112,y -< o(~ -k) II g II~ Repeating this argument for the eigenvalues near i and on the spectrum near O, one gets the result.

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. The existence (and uniqueness) a polynomial fellows from our previous considerations bation theory for ~ of such on the pertur- and from the invertibillty of the exponential function. , N. 21) Deferring the proof to Section 8, we shall now state the main estimate which leads to the existence of ~ ( x ) .

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