## 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

### MSC:

 60F10 Large deviations 60G60 Random fields
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