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
Giardina, C., & Redig, F. (2026). Duality for Markov processes: a Lie algebraic approach. Springer Nature.
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.
Junné, J., Redig, F., & Versendaal, R. (2024). Hydrodynamic limit of the symmetric exclusion process on complete Riemannian manifolds and principal bundles. arXiv:2410.20167.
Giardinà, C., Redig, F., & van Tol, B. (2024). Intertwining and propagation of mixtures for generalized KMP models and harmonic models. arXiv:2406.01160.
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.
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.
Frassek, R., & Giardinà, C. (2022). Exact solution of an integrable non-equilibrium particle system. Journal of Mathematical Physics, 63(10).
15:30– Europe/Lisbon Unusual schedule
Room P3.10, Mathematics Building Instituto Superior Técnico
— Online
From Statistical Mechanics to Quantum Mechanics, Quantum Field Theory or even Hydrodynamics, Probability has progressively become a fundamental toolbox in Mathematical Physics, unifying methods of research in domains that were regarded as separate for a long time. This series of seminars aims to showcase bridges between such domains, including those too recent to be regarded as associated with Theoretical or Mathematical Physics.
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