93-111 Fabio Martinelli , Enzo Olivieri

Some Remarks on Pathologies of Renormalization Group Transformations for the Ising Model (31K, plain TeX) May 7, 93

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Abstract. The results recently obtained by van Enter, Fernandez and Sokal [EFS] on non-Gibbsianness of the measure $\nu\,=\,T_b\,\mu_{\beta ,h}$ arising from the application of a single decimation transformation $T_b$, with spacing $b$, to the Gibbs measure $\mu_{\beta ,h}$ of the Ising model, for suitably chosen large inverse temperature $\beta$ and non zero external field $h$, are critically analyzed. In particular we show that if, keeping fixed the same values of $\beta$, $h$ and $b$, one iterates a sufficiently large number of times $n$ the transfomation $T_b$, one obtains a new measure $\nu '\,=\,(T_b)^n\,\mu_{\beta ,h}$ which is Gibbsian and moreover very weakly coupled.

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