Duality and hidden symmetries in interacting particle systems

TL;DR

This paper explores the relationship between duality functions and symmetries in Markov processes, revealing hidden symmetries in models like symmetric exclusion and KMP, with implications for interacting diffusions.

Contribution

It establishes a general framework linking duality and symmetries in Markov processes, including quantum spin representations and hidden SU(2) and SU(1,1) symmetries.

Findings

Identifies SU(2) symmetry in symmetric exclusion processes.

Unveils SU(1,1) symmetry in KMP model.

Shows all Brownian energy processes possess SU(1,1) symmetry.

Abstract

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.