Large Deviations for the Empirical Field of a Gibbs Measure

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 ∈ Zᵈ: 0 ≤ iₖ < n, 1 ≤ k ≤ 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⁻¹Vₙ, 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.