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