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2021 Consistent particle systems and duality
Gioia Carinci, Cristian Giardinà, Frank Redig
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Electron. J. Probab. 26: 1-31 (2021). DOI: 10.1214/21-EJP684


We consider consistent particle systems, which include independent random walkers, the symmetric exclusion and inclusion processes, as well as the dual of the Kipnis-Marchioro-Presutti model. Consistent systems are such that the distribution obtained by first evolving n particles and then removing a particle at random is the same as the one given by a random removal of a particle at the initial time followed by evolution of the remaining n1 particles.

In this paper we discuss two main results. Firstly, we show that, for reversible systems, the property of consistency is equivalent to self-duality, thus obtaining a novel probabilistic interpretation of the self-duality property. Secondly, we show that consistent particle systems satisfy a set of recursive equations. This recursions implies that factorial moments of a system with n particles are linked to those of a system with n1 particles, thus providing substantial information to study the dynamics. In particular, for a consistent system with absorption, the particle absorption probabilities satisfy universal recurrence relations.

Since particle systems with absorption are often dual to boundary-driven non-equilibrium systems, the consistency property implies recurrence relations for expectations of correlations in non-equilibrium steady states. We illustrate these relations with several examples.


We thank Mario Ayala and Federico Sau for useful discussions and suggestions.


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Gioia Carinci. Cristian Giardinà. Frank Redig. "Consistent particle systems and duality." Electron. J. Probab. 26 1 - 31, 2021.


Received: 8 October 2020; Accepted: 10 August 2021; Published: 2021
First available in Project Euclid: 5 October 2021

Digital Object Identifier: 10.1214/21-EJP684

Primary: 60J25 , 60K35

Keywords: boundary driven systems , Duality , interacting particle systems , non-equilibrium stationary measure , Symmetric exclusion process , symmetric inclusion process


Vol.26 • 2021
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