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This monograph studies duality in interacting particle systems, a topic combining probability theory, statistical physics, Lie algebras, and orthogonal polynomials. It offers the first comprehensive account of duality theory in the context of interacting particle systems.
Using a Lie algebraic framework, the book demonstrates how dualities arise in families of systems linked to algebraic representations. The exposition centers on three key processes: independent random walks, the inclusion process, and the exclusion process associated with the Heisenberg, su(1,1), and su(2) algebras, respectively. From these three basic cases, several new processes and their duality relations are derived. Additional models, such as the Brownian energy process, the KMP model and the Kac model, are also discussed, along with topics like the hydrodynamic limit and non-equilibrium behavior. Further, integrable systems associated to the su(1,1) algebra are studied and their non-equilibrium steady states are computed.
Intentionally accessible and self-contained, this book is aimed at graduate-level researchers and also serves as a comprehensive introduction to the duality of Markov processes and beyond.
Inhaltsverzeichnis
Chapter 1. Introduction.- Chapter 2. Basics of the algebraic approach.- Chapter 3. Duality for independent random walkers: part 1.- Chapter 4. Duality for independent random walkers: part 2.- Chapter 5. Duality for the symmetric inclusion process.- Chapter 6. Duality for the Brownian energy process.- Chapter 7. Duality for the symmetric partial exclusion process.- Chapter 8. Duality for other models.- Chapter 9. Orthogonal dualities.- Chapter 10. Consistency.- Chapter 11. Duality for non-equilibrium systems.- Chapter 12. Duality and macroscopic fields.- Chapter 13. Duality and integrable models.
Über den Autor / die Autorin
Cristian Giardinà is a professor of mathematics at Modena and Reggio Emilia University. His research activity centers on probability theory, statistical physics, and interacting particle systems. He has been working on duality properties of Markov processes, the hydrodynamic limit of open systems, large deviations and cloning algorithms, and the Ising model on random graphs. The overall aim is to understand how complex macroscopic behavior arises from microscopic models. He has published over 90 scientific papers, and two monographs on “Free boundary problems in PDE and particle systems” and “Perspectives on spin glasses”.
Frank Redig is a professor of probability theory at the Delft Institute of Applied Mathematics. His research focuses on Markov process theory, Gibbs measures, non-equilibrium statistical physics, and interacting particle systems.
Zusammenfassung
This monograph studies duality in interacting particle systems, a topic combining probability theory, statistical physics, Lie algebras, and orthogonal polynomials. It offers the first comprehensive account of duality theory in the context of interacting particle systems.
Using a Lie algebraic framework, the book demonstrates how dualities arise in families of systems linked to algebraic representations. The exposition centers on three key processes: independent random walks, the inclusion process, and the exclusion process—associated with the Heisenberg, su(1,1), and su(2) algebras, respectively. From these three basic cases, several new processes and their duality relations are derived. Additional models, such as the Brownian energy process, the KMP model and the Kac model, are also discussed, along with topics like the hydrodynamic limit and non-equilibrium behavior. Further, integrable systems associated to the su(1,1) algebra are studied and their non-equilibrium steady states are computed.
Intentionally accessible and self-contained, this book is aimed at graduate-level researchers and also serves as a comprehensive introduction to the duality of Markov processes and beyond.
