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Large Deviations for the Empirical Field of a Gibbs Measure

Hans Follmer and Steven Orey
The Annals of Probability
Vol. 16, No. 3 (Jul., 1988), pp. 961-977
Stable URL: http://www.jstor.org/stable/2244103
Page Count: 17
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Large Deviations for the Empirical Field of a Gibbs Measure
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Abstract

Let S be a finite set and consider the space Ω of all configurations ω: Zd → S. For j ∈ Zd, θj: Ω → Ω denotes the shift by j. Let Vn denote the cube $\{i \in Z^d: 0 \leq i_k < n, 1 \leq k \leq d\}$. Let μ be a stationary Gibbs measure for a stationary summable interaction. Define ρVn as the random probability measure on Ω given by ρVn (ω) = n-d ∑j ∈ Vn δθjω. Our principal result is that the sequence of measures μ ⚬ ρ-1 Vn , n = 1,2,⋯, satisfies the large deviation principle with normalization nd and rate function the specific relative entropy h(·; μ). 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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