## The Annals of Probability

- Ann. Probab.
- Volume 14, Number 2 (1986), 371-396.

### Duality for General Attractive Spin Systems with Applications in One Dimension

#### Abstract

A duality theory is developed which works for general Markovian spin-flip systems with attractive rates. This theory is applied to one-dimensional nearest neighbor translation invariant systems to extend results which were first proved for the contact process by Durrett and Griffeath (1983). In particular, exponential convergence to equilibrium starting from all 1's is shown for noncritical nonergodic systems (Theorem 2). As a consequence, two different definitions of the critical value are shown to be equivalent (Theorem 5). In the course of the proof of Theorem 2, a new result concerning the distribution of the system near edges is obtained (Theorem 4).

#### Article information

**Source**

Ann. Probab., Volume 14, Number 2 (1986), 371-396.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176992522

**Digital Object Identifier**

doi:10.1214/aop/1176992522

**Mathematical Reviews number (MathSciNet)**

MR832015

**Zentralblatt MATH identifier**

0604.60098

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

**Keywords**

Duality spin system exponential mixing graphical methods percolation

#### Citation

Gray, Lawrence. Duality for General Attractive Spin Systems with Applications in One Dimension. Ann. Probab. 14 (1986), no. 2, 371--396. doi:10.1214/aop/1176992522. https://projecteuclid.org/euclid.aop/1176992522