Orateur
Description
Duality is a powerful tool in the study of large-scale stochastic systems (interacting particle systems). These interacting particle systems are used to model phenomena in non-equilibrium statistical physics, as well as in mathematical population genetics. Duality allows to relate complicated interacting systems to simpler systems, e.g. systems with only a few particles, or systems where non-equilibrium driving has been replaced by absorbing boundaries.
In the last decade, a lot of work has been done to relate duality of Markov processes to symmetries of the underlying Markov generator, and from there one is naturally lead towards simple objects in Lie representation theory such as the Casimir, coproduct, tensor products, which become tools to constructively find processes with useful dualities. For symmetric processes (detailed balance setting) these Lie algebras are classical ones (such as SU(2), SU(1,1), SU(n), SU(n,n)), whereas for the corresponding asymmetric processes, their quantum deformations are used.
In this seminar, starting with simple examples, we show how the Lie-algebraic approach allows to unify several processes, and provides a systematic method to produce processes which have some degree of exact solvability. We start with the classical processes such as independent random walkers and the symmetric inclusion process. We will also discuss recent developments in the representation of non-equilibrium states as mixtures of product measures, and how this can be used to study fluctuations of the density profile for such systems.