Abstract
We consider real-space renormalization group transformations for Ising-type systems which are formally defined by
whereT(σ, σ′) is a probability kernel, i.e., ∑σ′ T(σ,σ′) = 1 for every configuration σ. For each choice of the block spin configuration σ′, let σ′, let μσ′ be the measure on spin configurations σ which is formally given by taking the probability of σ to be proportional toT(σ, σ′) exp[−H(σ)]. We give a condition which is sufficient to imply that the renormalized HamiltonianH′ is defined. Roughly speaking, the condition is that the collection of measures μσ′ is in the high-temperature phase uniformly in the block spin configuration σ′. The proof of this result uses methods of Olivieri and Picco. We use our theorem to prove that the first iteration of the renormalization group transformation is defined in the following two examples: decimation with spacingb = 2 on the square lattice with β < 1.36β c and the Kadanoff transformation with parameterp on the trian gular lattice in a subset of the β,p plane that includes values of β greater than β c .
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Haller, K., Kennedy, T. Absence of renormalization group pathologies near the critical temperature. Two examples. J Stat Phys 85, 607–637 (1996). https://doi.org/10.1007/BF02199358
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DOI: https://doi.org/10.1007/BF02199358