## The Annals of Probability

### 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.

#### Article information

Source
Ann. Probab., Volume 16, Number 3 (1988), 961-977.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176991671

Digital Object Identifier
doi:10.1214/aop/1176991671

Mathematical Reviews number (MathSciNet)
MR942749

Zentralblatt MATH identifier
0648.60028

JSTOR