WIAS Preprint No. 606, (2000)

On the Gibbsian nature of the random field Kac model under block-averaging


  • Külske, Christof

2010 Mathematics Subject Classification

  • 82B44 82B28 82B20


  • Kac-model, random field model, Gibbs-measures, renormalization group transformations


We consider the Kac-Ising model in an arbitrary configuration of local magnetic fields η = (ηi)i∈𝐙d, 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.

Appeared in

  • J. statist. Phys. 104 (2001), pp. 991-1012

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