Open Access
April, 1986 Duality for General Attractive Spin Systems with Applications in One Dimension
Lawrence Gray
Ann. Probab. 14(2): 371-396 (April, 1986). DOI: 10.1214/aop/1176992522


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).


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Lawrence Gray. "Duality for General Attractive Spin Systems with Applications in One Dimension." Ann. Probab. 14 (2) 371 - 396, April, 1986.


Published: April, 1986
First available in Project Euclid: 19 April 2007

zbMATH: 0604.60098
MathSciNet: MR832015
Digital Object Identifier: 10.1214/aop/1176992522

Primary: 60K35

Keywords: Duality , Exponential mixing , graphical methods , percolation , Spin system

Rights: Copyright © 1986 Institute of Mathematical Statistics

Vol.14 • No. 2 • April, 1986
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