By Pierre Collet
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Extra resources for A Renormalization Group Analysis of the Hierarchical Model in Statistical Mechanics
Is a variant of an argument suggested by Nappi-Hegerfeldt and given in  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. Then the fixed point ~ are standard methods is in L and there to discuss the flow induced by the map ~ ~(~ On Banach spaces, + %) - ~(%) =: T(~) .
In fact one checks, = 6(6 - I) , Another critical index, < s o sj>2N f = i~i f for all c . "0", describes the behavlour of dsi 2N H ds i i=i SoSj exp(-6~2N, f) exp(- 6 ~2N, f ) , of 58 as a function of J at the critical temperature does not depend on any eigenvalue. Due to the particular of the model, definition we have for J < N < SoS2J > 2 N f c < : < (So+Sl)/2 ' (s2J + s2J+1)/2 >N 2,f So s2J-I > 2N-I' JT~ S)(f) In the scaling limit, < as j ~ ~ . This index we fix N' = N-J : SoS2J > 2N'+J f ' ~ = 8crit " and write c-J < So 2 > 2 N', ~c~)J(f) .
Y ' (of. 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.
A Renormalization Group Analysis of the Hierarchical Model in Statistical Mechanics by Pierre Collet