By Pierre Collet

ISBN-10: 3540086706

ISBN-13: 9783540086703

ISBN-10: 3540358994

ISBN-13: 9783540358992

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**Extra resources for A Renormalization Group Analysis of the Hierarchical Model in Statistical Mechanics**

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Indeed, log a~ 1, 86 1N ~, _ _ + a2~2 r~ ~ . But 0 if D ~ O, as the expectation of S in an odd measure, and it is bounded by continuity. 33) remains true if N ll BN hN = exp(-g(N)) with g(N) > O, g(N) ~ 0 as N ~ N Assume now, as befere, M~,h,f = that the thermodynamic N lira -~ o~ MN, ~,h, f limit 56 exists. 29) carries over and we get log X2 . 32) is also equal to log M P 2N' ~N' hN# f lim N ~ l o g hN ~ l o g Z2 / l o g ~'l i and k i = 2c -~ . This describes the magnetization perature as a function of the magnetic Finally assume that M~,f at the critical tem- field.

30) -N according to the principles we have mtated in the case of the free energy. )) lim N -~ ~ log I~N - 0critl + log MLI,6N•aI~IXN6NhN + N . 32) log k 2 if log M~I•~N,al@lkN6NhN+ a2e2k26 N as N ~ ~ . s. 31) hold. Indeed, log a~ 1, 86 1N ~, _ _ + a2~2 r~ ~ . But 0 if D ~ O, as the expectation of S in an odd measure, and it is bounded by continuity. 33) remains true if N ll BN hN = exp(-g(N)) with g(N) > O, g(N) ~ 0 as N ~ N Assume now, as befere, M~,h,f = that the thermodynamic N lira -~ o~ MN, ~,h, f limit 56 exists.

35) near the critical in the two phase region. Summarizing, near the critical we see that the behaviour of the various quantities temperature completely controlled or near the Critical by the llnearization field (h = O) is of the tangent m a p ~ j ~ ( ~ c ). 57 Since we have seen that ~c is C ~ analytically in e ~ 0, and since 9J~e(e~) depends on e~ , the (isolated) elgenvalues of ~J~c(~) have asymp- I totic expansions (actually, k ° = 2, k I = 2/c w , so that this statement Is only relevant for k2).

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

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