Large Deviations for the Empirical Field of a Gibbs Measure

Abstract

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.