The Annals of Probability

On the Uniqueness and Nonuniqueness of Proximity Processes

Lawrence Gray and David Griffeath

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Abstract

We discuss the uniqueness of a class of infinite particle systems known as proximity processes, with the aid of certain "dual" Markov chains. By checking whether the dual "explodes," i.e., attempts infinitely many jumps in a finite time, and how it explodes when it does, it is possible in many cases to determine whether or not there is more than one particle system with given flip rates. We then use duality to find an example of a system which is uniquely determined by its flip rates, but whose generator is not the closure of the naive operator formed from these flip rates.

Article information

Source
Ann. Probab., Volume 5, Number 5 (1977), 678-692.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176995712

Digital Object Identifier
doi:10.1214/aop/1176995712

Mathematical Reviews number (MathSciNet)
MR448629

Zentralblatt MATH identifier
0411.60099

JSTOR
links.jstor.org

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

Keywords
Infinite particle system uniqueness nonuniqueness

Citation

Gray, Lawrence; Griffeath, David. On the Uniqueness and Nonuniqueness of Proximity Processes. Ann. Probab. 5 (1977), no. 5, 678--692. doi:10.1214/aop/1176995712. https://projecteuclid.org/euclid.aop/1176995712


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