Large deviations for the empirical field of a Gibbs measure. (English) Zbl 0648.60028

Let S be a finite set and \(\Omega\) the set of configurations \(\omega: {\mathbb{Z}}^ d\to S.\) For \(j\in {\mathbb{Z}}^ d\), \(\theta_ j: \Omega \to \Omega\) denotes the shift by j. Let \(V_ n\) denote the cube \(\{i\in {\mathbb{Z}}^ d:\) \(0\leq i_ k<n\), \(1\leq k\leq d\}\). Let \(\mu\) be a stationary Gibbs measure for a stationary summable interaction. Define \(\rho_{V_ n}(\omega)=n^{-d}\sum_{j\in V_ n}\delta_{\theta_ j\omega}.\)
The authors show that the sequence of measures \(\mu \circ \rho^{- 1}_{V_ n}\) satisfies a large deviation principle with normalization \(n^ d\) and the specific relative entropy h(.;\(\mu)\) as rate function. Applying the contraction principle, they obtain a large deviation principle for the distribution of the empirical distributions.
Reviewer: M.Scheutzow


60F10 Large deviations
60G60 Random fields
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