# Lectures on Duality for interacting particle systems: the algebraic approach, via examples, by Frank Redig (CPHT, Chaire d’Alembert, 1 January-31 June 2020)

**Frank Redig will give six lectures, grouped into three sessions, at the following dates : 19 May, 26 May, 2 June, from 10:30 to 12:30, in the conference room Louis Michel.**

The lectures will be centred around the theory and applications of duality in the context of interacting particle systems and non-equilibrium statistical physics models. Duality is a powerful tool in Markov process theory, allowing to connect two processes (the process under study and its dual) via a so-called duality function. Often the dual process is simpler, e.g., in systems coupled with reservoirs in the dual the reservoirs are replaced by absorbing boundaries, or in infinite interacting systems, the dual is a system of finitely many particles, or in the context of diffusion processes, the dual is a simple discrete jump process.

We show how to understand dualities from the point of view of an underlying Lie algebra of symmetries (operators commuting with the generator).

This approach gives several new dualities, and new duality functions, such as orthogonal duality functions. It also allows to constructively define processes with dualities, and to find ``correct’’ asymmetric versions of symmetric models with dualities (via so-called q-deformation).

We will explain this theory starting from simple examples (such as independent random walkers, exclusion process, inclusion process), and provide several applications in the description of non-equilibrium steady states as well as in hydrodynamic limits and fluctuation fields.

**Lecture 1**: Introduction, motivation, some background from Markov process theory

**Lecture 2**: Duality: definition, duality and symmetries, duality and intertwining, duality and change of representation. Examples.

**Lecture 3**: Dualities for independent random walkers: self-duality, duality with deterministic system, averaging models. Applications to hydrodynamic limits and fluctuation fields and to ergodic theory.

**Lectures 4-5**: The symmetric inclusion process (SIP) and its dual processes: Lie algebraic construction via co-product of the Casimir.

Discrete and continuous representations, diffusion processes dual to SIP. Thermalization, models of KMP type, inhomogeneous systems.

**Lecture 6**: Non-equilibrium steady states: duality with reservoirs. Consistency, orthogonal polynomial duality. Universal properties of correlation functions of non-equilibrium steady states.