In this talk we will introduce the concept of duality for a class of interacting particle systems that includes the well-known symmetric inclusion and exclusion processes. By means of examples we will show how to derive self-duality relations from the knowledge of reversible measures, using the so-called algebraic approach. We will also show how to derive orthogonal dualities by means of the well known three terms recurrence relations satisfied by the classical (discrete) families of orthogonal polynomials.
As a first application, we will use duality to show the propagation of positive correlations for one of our processes. Later we will also provide applications of (self-)duality in the study of macroscopic fields of interacting particle systems, in particular in the derivation of hydrodynamic limits and fluctuations from the hydrodynamic limit. Finally, if time permits, we will try to make the case (via examples) that interacting particle systems that enjoy the property of self-duality can be used as a sort of laboratory to test the validity of general claims about interacting particle systems.
Disclaimer: This material is mostly based on the work of others (to mention a few: C. Franceschini, G. Carinci, C. Giardina, and F. Redig). However, some of the applications of orthogonal duality are based on the speaker's own work together with Gioia Carinci and Frank Redig.