Here 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.

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).

Here 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.

Here 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.

We present a unified perspective on the hydrodynamic limits of three interacting particle systems in contact with slow boundary reservoirs: the Simple Symmetric Exclusion Process (SSEP), the Porous Medium Model (PMM), and the Symmetric Zero-Range Process (ZR).

Although these systems share the same type of boundary dynamics — particle creation and annihilation at rates of order $N^{-\theta}$ — their bulk dynamics differ substantially: linear exclusion, constrained exclusion with nonlinear mobility, and unbounded occupancy with nonlinear jump rates.

Under diffusive scaling, the empirical density evolves according to a parabolic equation whose form depends on the microscopic interaction. We show how the strength of the reservoirs determines a phase transition in the macroscopic boundary conditions: Dirichlet for $\theta < 1$, Robin for $\theta = 1$, and Neumann for $\theta > 1$.

This comparison highlights how microscopic mechanisms shape macroscopic diffusion, while revealing a universal boundary transition driven by slow reservoirs.

Many complex biological and physical networks are naturally subject to both random influences, i.e., extrinsic randomness, from their surrounding environment, and uncertainties, i.e., intrinsic noise, from their individuals. Among many interesting network dynamics, of particular importance is the synchronization property which is closely related to the network reliability especially in cellular bio-networks. It has been speculated that whereas extrinsic randomness may cause noise-induced synchronization, intrinsic noises can drive synchronized individuals apart. This talk presents an appropriate framework of (discrete-state and discrete time) Markov random networks to incorporate both extrinsic randomness and intrinsic noise into the rigorous study of such synchronization and desynchronization scenaria. In particular, alternating patterns between synchronization and desynchronization behaviors are given by studying the asymptotics of the Markov perturbed stationary distributions. This talk is based on joint works with Arno Berger, Wen Huang, Hong Qian, Felix X.-F. Ye, and Yingfei Yi.