Large Deviations for the Empirical Field of a Gibbs Measure

Let S be a finite set and consider the space of all configurations: Zᵈ S. For j Zᵈ, ⱼ: denotes the shift by j. Let Vₙ denote the cube \i Zᵈ: 0 iₖ < n, 1 k d\. Let be a stationary Gibbs measure for a stationary summable interaction. Define ₕ䂸 as the random probability measure on given by ₕ䂸 () = n^-d ₉ ₕ䂸 䲛. Our principal result is that the sequence of measures ^-1ₕ䂸, n = 1, 2, , satisfies the large deviation principle with normalization nᵈ and rate function the specific relative entropy h (;). 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.