A Renormalization Group Analysis of the Hierarchical Model by Pierre Collet PDF

By Pierre Collet

ISBN-10: 3540086706

ISBN-13: 9783540086703

ISBN-10: 3540358994

ISBN-13: 9783540358992

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Additional info for A Renormalization Group Analysis of the Hierarchical Model in Statistical Mechanics

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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(~) . 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.

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. 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.

1). The Case I above corresponds to the critical temperature and the thermodynamic limit does not exist. However, differently rescaled va- riables could be made to converge. Ne now discuss Case 2 in more detail. We first give a collection of more intuitive facts which are of relevance for the mathematical proof of Part II. i) The "function" ~ is a fixed point of IT. Indeed, z ~-~ z ~-~ / e_U 2 6(zc-½ + U) 6(ze -½ - U) du z 6(2c-~z)~(-z2/c) = z ~-~- i e~ ~(Z)->-- , 62 l so that 6(x) • ~ i 2c -~ is a fixed point of J~.

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

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