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July, 1988 Large Deviations for the Empirical Field of a Gibbs Measure
Hans Follmer, Steven Orey
Ann. Probab. 16(3): 961-977 (July, 1988). DOI: 10.1214/aop/1176991671


Let $S$ be a finite set and consider the space $\Omega$ of all configurations $\omega: Z^d \rightarrow S$. For $j \in Z^d, \theta_j: \Omega \rightarrow \Omega$ denotes the shift by $j$. Let $V_n$ denote the cube $\{i \in Z^d: 0 \leq i_k < n, 1 \leq k \leq d\}$. Let $\mu$ be a stationary Gibbs measure for a stationary summable interaction. Define $\rho_{V_n}$ as the random probability measure on $\Omega$ given by $\rho_{V_n}(\omega) = n^{-d} \sum_{j \in V_n} \delta_{\theta_j\omega}.$ Our principal result is that the sequence of measures $\mu \circ \rho^{-1}_{V_n}, n = 1,2,\cdots$, satisfies the large deviation principle with normalization $n^d$ and rate function the specific relative entropy $h(\cdot; \mu)$. Applying the contraction principle, we obtain a large deviation principle for the distribution of the empirical distributions; a detailed description of the resulting rate function is provided.


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Hans Follmer. Steven Orey. "Large Deviations for the Empirical Field of a Gibbs Measure." Ann. Probab. 16 (3) 961 - 977, July, 1988.


Published: July, 1988
First available in Project Euclid: 19 April 2007

zbMATH: 0648.60028
MathSciNet: MR942749
Digital Object Identifier: 10.1214/aop/1176991671

Primary: 60F10
Secondary: 60G60

Keywords: Entropy , Gibbs measures , large deviations , Random fields

Rights: Copyright © 1988 Institute of Mathematical Statistics

Vol.16 • No. 3 • July, 1988
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