Duality and hidden symmetries in interacting particle systems

In the context of Markov processes, both in discrete and continuous setting, we show a general relation between duality functions and symmetries of the generator. If the generator can be written in the form of a Hamiltonian of a quantum spin system, then the “hidden” symmetries are easily derived. We illustrate our approach in processes of symmetric exclusion type, in which the symmetry is of SU(2) type, as well as for the Kipnis-Marchioro-Presutti (KMP) model for which we unveil its SU(1, 1) symmetry. The KMP model is in turn an instantaneous thermalization limit of the energy process associated to a large family of models of interacting diffusions, which we call Brownian energy process (BEP) and which all possess the SU(1, 1) symmetry. We treat in details the case where the system is in contact with reservoirs and the dual process becomes absorbing. Department of Mathematics and Computer Science, Eindhoven University, P.O. Box 513 5600 MB Eindhoven, The Netherlands, [email protected] CNRS-Ecole Supérieure de Physique et de Chimie Industrielles , rue Vauquelin 10, 75231 Paris, France, [email protected] Mathematisch Instituut Universiteit Leiden, Niels Bohrweg 1, 2333 CA Leiden, The Netherlands, [email protected] Mathematisch Instituut Universiteit Leiden, Niels Bohrweg 1, 2333 CA Leiden, The Netherlands, [email protected]