Summary
This paper contains mainly an exposition of Wilson’s theory of the Kadanoff renormalization group.
Riassunto
Questo lavoro contiene essenzialmente un’esposizione della teoria di Wilson del gruppo di rinormalizzazione di Kadanoff.
Резюме
В этой работе рассматривается теория Вильсона для группы перенормировки Каданова.
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References
P. H. Bleher andJa. G. Sinai:Comm. Math. Phys.,33, 23 (1973).
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Hence if we consider the Ising model\(\left( {\beta _c \bar H_0 ,\bar \pi _0 } \right)\) at the critical temperature, the models\(\left( {\beta _c \bar H_0 ,\bar \pi _0 } \right)\) will be in some «neighbourhood» only ifβ⩽β c.
There is a very nice counterexample due toKasteleyn to the general validity of such a statement: consider the two-dimensional Ising model with the constraint that the block spins {x, 1} are all zero and furthermore the block configurations ±±and±±are forbidden; this model is exactly solvable and has the same critical temperature as the original unconstrained Ising model. (Kasteleyn: private communication.)
I. I. Ghichman andA. V. Scorochod:Theory of Random Processes (Moscow, 1971);P. L. Dobrushin:Theory of prob. and applications,13, 197 (1968);25, 458 (1970).
The existence of the above limit in a sense stronger than the weak one just described can be rigorously established for noncritical cases cf. ref. (6)G. Gallavotti, H. J. F. Knops andG. C. Martin Löf: preprint (1974).
The existence of a γ such that (4.1) holds is essentially equivalent to the requirement that the pair correlation function in (H, π) behaves as a pure power law for large distances: if<σ0σ R >≃R −(d−2+n), then γ=1+(2-η)/d. If this is not true,i.e. there are, say, logarithmic corrections, then one has to choose θ as a function ofn but, usually,\(\mathop {\lim }\limits_{x \to \infty } \rho _n \) will exist and one could try to give arguments similar to the ones used in the case of constant θ. We do not enter here in this discussion.
K. G. Wilson andM. E. Fisher:Phys. Rev. Lett.,28, 240 (1972);M. E. Fisher:Phys. Rev. Lett.,29, 917 (1972).
G. Gallavotti, H. J. F. Knops andH. van Beyeren: preprint (1974).
This would be more natural if the constrained Hamiltonian(H l ,π l )were never critical or had, in some sense, an ∞-order transition (cf. Sect.4, discussion of assumption I)).
F. J. Dyson:Comm. Math. Phys.,12, 91, 212 (1969);21, 269 (1971).
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Ja. G. Sinai: preprint.
Notice that ifB(v,v)is a bilinear form in thev l 's a sufficient condition for (7.6) is a given by\(\sum\limits_{J = 1}^{2^d } {B_{lJ} } = \propto _0 \)(independent ofi!), which is a condition of symmetry saying that all the spins in the same block enter symmetrically in the Hamilktoinian. The conditions on the signs of the eigenvalues ofB is a kind of attractiveness condition which is needed also to avoid divergences in (7.8).
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«We must lay it down as an incontestable axiom, that in all operations of art and nature, nothing is created: an equal quantity of matter exists both before and after the experiment … and nothing takes place beyond changes and modifications of these elements».A. L. Lavoisier:Traité élèmentaire de chimie préssenté dans un ordre nouveau (Paris, 1789) (new edition, Paris, 1937).
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Cassandro, M., Gallavotti, G. The Lavoisier law and the critical point. Nuov Cim B 25, 691–705 (1975). https://doi.org/10.1007/BF02724745
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DOI: https://doi.org/10.1007/BF02724745