Special sessions

A three day short course (28, 29 and 30/4/2026 at P3.10@Técnico and online) by

Frank Redig, TUDelft
Particle systems and duality

In the first lecture we introduce basic concepts, various models (SIP, SEP, independent random walkers) and how they are linked to each other via the Lie algebraic formalism. From the Lie algebraic formalism we infer that interacting particle systems with dualities come in families characterized by an underlying Lie algebra. These are SU(2) for SEP, SU(1,1) for SIP, and the Heisenberg algebra for independent particles.

In the second lecture we use duality to characterize the ergodic invariant measures, and use duality to also look at the stationary state of systems driven by reservoirs at the boundary. Special attention is given to the harmonic model and propagation of mixed product states.

Finally, in the last lecture, we use duality to characterize hydrodynamic limits and fluctuation fields. Special attention is given to the hydrodynamic limit of SEP in a geometric setting, i.e., on graphs that approximate a Riemannian manifold.

References

1. Giardina, C., & Redig, F. (2026). Duality for Markov processes: a Lie algebraic approach. Springer Nature.
2. Van Ginkel, B., & Redig, F. (2020). Hydrodynamic Limit of the Symmetric Exclusion Process on a Compact Riemannian Manifold: B. van Ginkel et al. Journal of Statistical Physics, 178(1), 75-116.
3. Junné, J., Redig, F., & Versendaal, R. (2024). Hydrodynamic limit of the symmetric exclusion process on complete Riemannian manifolds and principal bundles. arXiv:2410.20167.
4. Giardinà, C., Redig, F., & van Tol, B. (2024). Intertwining and propagation of mixtures for generalized KMP models and harmonic models. arXiv:2406.01160.
5. Schütz, G., & Sandow, S. (1994). Non-Abelian symmetries of stochastic processes: Derivation of correlation functions for random-vertex models and disordered-interacting-particle systems. Physical Review E, 49(4), 2726.
6. Giardina, C., Kurchan, J., Redig, F., & Vafayi, K. (2009). Duality and hidden symmetries in interacting particle systems. Journal of Statistical Physics, 135(1), 25-55.
7. Frassek, R., & Giardinà, C. (2022). Exact solution of an integrable non-equilibrium particle system. Journal of Mathematical Physics, 63(10).

Permanent link to this information: https://pmp.math.tecnico.ulisboa.pt/lecture_series?sgid=114