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Kadanoff. A measure µ ∈ P(Ω) is called self-similar if it is a fixed point of Rκ,ζ , that is, it is a solution of Rκ,ζ (µ) = µ. (114) The basic idea of the renormalization group theory of critical phenomena is the so-called universality hypothesis, which states that at the critical point Mathematical Theory of the Ising Model and its Generalizations 45 β = βc , the set of all Gibbs states of the model Gβ consists of one element and this element is self-similar. In other words, at the critical point the individual spins have the same probability distribution as the renormalized sums of such spins, as well as the renormalized sums of these sums, and so on.

G. Sinai in Ref. 81, was formulated as a property of a random field. Later the concept of self-similar random fields became a part of the mathematical theory of critical phenomena in models of statistical physics, lattice spin models in particular. A full and comprehensive description of this concept is given in Refs. 14 and 66. In this connection we mention also papers by K. 82−84 Renormalization group methods developed in theoretical physics have produced and still are producing a very strong impact on the theory of critical phenomena.

Given such a cube Λ, we define |l − l |Λ = |l1 − l1 |2Λ + · · · + |ld − ld |2Λ , where |lj − lj |Λ = min{|lj − lj |; 2L − |lj − lj |}, j = 1, . . , d. The above introduced function gives a periodic distance between the points l, l ∈ Λ, it may be interpreted as a distance on the torus which one obtains by identifying the opposite walls of the cube Λ. Thereby, we set JllΛ = φ(|l − l |Λ ) with the same φ. The Hamiltonian HΛ = − 1 2 JllΛ σl σl − l,l ∈Λ hσl , (66) l∈Λ is invariant with respect to translations on the mentioned torus.

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On the estimation of distances in a curved universe with a non-static line element by Tolman R.C.


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