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

Abstract

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.