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On the Gibbsian Nature of the Random Field Kac Model Under Block-Averaging

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Abstract

We consider the Kac–Ising model in an arbitrary configuration of local magnetic fields η = \(\left( {\eta _{{\kern 1pt} i} } \right)_{i\;\quad \quad \in \quad {\kern 1pt} \quad \mathbb{Z}^d }\), in any dimension d, at any inverse temperature. We investigate the Gibbs properties of the 'renormalized' infinite volume measures obtained by block averaging any of the Gibbs-measures corresponding to fixed η, with block-length small enough compared to the range of the Kac-interaction. We show that these measures are Gibbs measures for the same renormalized interaction potential. This potential depends locally on the field configuration η and decays exponentially, uniformly in η, for which we give explicit bounds. The construction of the potential is based on a high temperature-type cluster expansion.

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  1. WIAS, Mohrenstrasse 39, D-10117, Berlin, Germany

    Christof Külske

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Külske, C. On the Gibbsian Nature of the Random Field Kac Model Under Block-Averaging. Journal of Statistical Physics 104, 991–1012 (2001). https://doi.org/10.1023/A:1010497510308

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