A Renormalization Group Analysis of the Hierarchical Model - download pdf or read online

By Pierre Collet

ISBN-10: 3540086706

ISBN-13: 9783540086703

ISBN-10: 3540358994

ISBN-13: 9783540358992

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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.

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. As we shall see below, the numbers v Q depend on ~ , and are called the "trivial indices" or "mean field indices" for s = O.

21) Deferring the proof to Section 8, we shall now state the main estimate which leads to the existence of ~ ( x ) . 30 Write m~(x) = fN(~,x) + R (x). 22) Using the quadratic nature of J~c' and the definition of ~2J~(c(c),~) ~ = 2 J~c(~,~' ) cf. 13) , we set ~$,e = ~ = ~2J~{c(~),~). 6. For all N > 0, there is an ~o(N) > 0 such that for 0 < e < ~o(N) the operator (~f with norm less than C N C 12. 22) posses- ses thus a unique fixed point. We have thus shown the existence of ~ . ) + O(e (N-i)/2) in L .

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A Renormalization Group Analysis of the Hierarchical Model in Statistical Mechanics by Pierre Collet


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