Consistent particle systems and duality

Gioia Carinci, Cristian Giardinà, Frank Redig

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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 n − 1 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 n − 1 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 expec-tations of correlations in non-equilibrium steady states. We illustrate these relations with several examples.

Original languageEnglish
Article number125
Pages (from-to)1-31
JournalElectronic Journal of Probability
Publication statusPublished - 2021


  • Boundary driven systems
  • Duality
  • Interacting particle systems
  • Non-equilibrium stationary measure
  • Symmetric exclusion process
  • Symmetric inclusion process

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