The Annals of Probability

Quantum operators in classical probability theory. IV. Quasi-duality and thinnings of interacting particle systems

Peter Lloyd and Aidan Sudbury

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Duality has proved to be a powerful technique in the study of interacting particle systems (IPS). This concept can be enlarged and a “quasi-duality” defined between various pairs of IPS previously thought unrelated. Consequently, theorems of a similar style to those involving duality can be deduced.

The concept of quasi-duality follows naturally from our previous studies into the use of “single-site operators” (an idea borrowed from quantum physics) in paper II of this series. It is shown that a necessary condition for quasi-duality is that the eigenvalues of the corresponding two-site infinitesimal generators be the same, and, using this observation, a number of quasi-dual pairs have been found and studied.

It is further shown that if two different IPS share a common dual, then one can be considered as a “thinning” of the other.

Article information

Ann. Probab., Volume 25, Number 1 (1997), 96-114.

First available in Project Euclid: 18 June 2002

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Zentralblatt MATH identifier

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

Infinite particle system duality


Sudbury, Aidan; Lloyd, Peter. Quantum operators in classical probability theory. IV. Quasi-duality and thinnings of interacting particle systems. Ann. Probab. 25 (1997), no. 1, 96--114. doi:10.1214/aop/1024404280.

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