## The Annals of Probability

- Ann. Probab.
- Volume 21, Number 4 (1993), 1845-1875.

### Large Deviations and Maximum Entropy Principle for Interacting Random Fields on $\mathbb{Z}^d$

#### Abstract

We present a new approach to the principle of large deviations for the empirical field of a Gibbsian random field on the integer lattice $\mathbb{Z}^d$. This approach has two main features. First, we can replace the traditional weak topology by the finer topology of convergence of cylinder probabilities, and thus obtain estimates which are finer and more widely applicable. Second, we obtain as an immediate consequence a limit theorem for conditional distributions under conditions on the empirical field, the limits being those predicted by the maximum entropy principle. This result implies a general version of the equivalence of Gibbs ensembles, stating that every microcanonical limiting state is a grand canonical equilibrium state. We also prove a converse to the last statement, and discuss some applications.

#### Article information

**Source**

Ann. Probab., Volume 21, Number 4 (1993), 1845-1875.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176989002

**Digital Object Identifier**

doi:10.1214/aop/1176989002

**Mathematical Reviews number (MathSciNet)**

MR1245292

**Zentralblatt MATH identifier**

0790.60031

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F10: Large deviations

Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B05: Classical equilibrium statistical mechanics (general)

**Keywords**

Large deviations maximum entropy principle Gibbs measure equilibrium state conditional limit theorem equivalence of ensembles microcanonical distribution empirical distribution

#### Citation

Georgii, Hans-Otto. Large Deviations and Maximum Entropy Principle for Interacting Random Fields on $\mathbb{Z}^d$. Ann. Probab. 21 (1993), no. 4, 1845--1875. doi:10.1214/aop/1176989002. https://projecteuclid.org/euclid.aop/1176989002