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Absence of renormalization group pathologies in some critical Dyson-Ising ferromagnets

The Dyson-Ising ferromagnet is a one-dimensional Ising model with a power law interaction. When the power is between -1 and -2, the model has a phase transition. Van Enter and Le Ny proved that at sufficiently low temperatures the decimation renormalization group transformation is not defined in the sense that the renormalized measure is not a Gibbs measure. We consider a modified model in which the nearest neighbor couplings are much larger than the other couplings. For a family of Hamiltonians which includes critical cases, we prove that the first step of the renormalization group transformation can be rigorously defined for majority rule and decimation.