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Almost Gibbsian versus weakly Gibbsian measures. (English) Zbl 0963.60094

Except for the standard Gibbs measure, two weaker notions are investigated. A probability measure \(\mu \) on \(\Omega =\{1,\dots ,q\}^{\mathbb Z^d}\) is almost Gibbsian if it corresponds to some uniformly positive specification \(\Gamma =\{\gamma _{\Lambda}(\cdot \mid \omega)\}\) such that for \(\mu \)-almost all \(\omega \in \Omega \) the kernels \(\gamma _{\Lambda}\) are continuous at \(\omega \). A probability \(\mu \) on \(\Omega \) is weakly Gibbsian if it corresponds to a potential \(\{U(A,\omega)\); \(A\subset \mathbb Z^d\) finite, \(\omega \in \Omega \}\) that is absolutely convergent for \(\mu \)-almost all \(\omega \). Each almost Gibbsian measure is weakly Gibbsian. This is derived from a pointwise version of a theorem of Kozlov. In terms of a set of “bad” configurations \(\xi \in \Omega \) for \(\mu \), Gibbsian measures and almost Gibbsian measures are characterized. Examples of non weakly Gibbsian and of weakly Gibbsian but not almost Gibbsian measures are given.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60G60 Random fields
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
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