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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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2 O 1 2 I/3 21/2 2 where each branch is controlled near the branching point. These branches correspond to "critical", "tricrltlcal", "tetrac~itical" ... behaviour, 34 as c = 2 I/2, 2 I/3 , 2 I/4 ... respectively. calculations Bleher ~ has done numerical and followed the c r i t i c a l b r a n c h almost to c = i . He feund no further bifurcations, the f u n c t i o n ~c behaves and the following diagram shows how (as a f u n c t i o n of c). The H i e r a r c h i c a l Model has no phase t r a n s i t i o n at c = I [i0].

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 . Note that this is only existence for c > 0, so we do not have a bifurcation into two solutions as would be implied by perturbation theory alone. 6 , because it should have some interest of its own. First we observe that 31 A£1g c has two distinct important features. i) ~4fN,e has an integral kernel which decays like 1 exp(-(zc -g - u) 2 -$~(2zc -½ - u) 4) and from this we get the bound if p ~> 2~c -I (of.

Then the fixed point ~ are standard methods is in L and there to discuss the flow induced by the map ~ ~(~ On Banach spaces, + %) - ~(%) =: T(~) . 1) the flow around a fixed point can be almost completely characterized by the tangent map at the fixed point. In our case T(%)(~) = cf. page ~(%)(~) = 2s (~,~) . g. i n L 2 , 7 , c~a = 1 - c ~ ' the previous i H4(x(l-2-g)~) + O(c 2) results of section. 1 . ) on L ficiently by the theory , consists, for suf- small ~ > 0 (depending on N) of eigenvalues j~4, 2/2 j / 4 + O(s) for j = O, ....

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

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