## The Annals of Probability

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

### 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**

links.jstor.org

**Subjects**

Primary: 60F10: Large deviations

Secondary: 60G60: Random fields

**Keywords**

Large deviations random fields Gibbs measures entropy

#### Citation

Follmer, Hans; Orey, Steven. Large Deviations for the Empirical Field of a Gibbs Measure. Ann. Probab. 16 (1988), no. 3, 961--977. doi:10.1214/aop/1176991671. https://projecteuclid.org/euclid.aop/1176991671