The Annals of Probability

Duality for General Attractive Spin Systems with Applications in One Dimension

Lawrence Gray

Full-text: Open access

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


Export citation