Interacting particle system

In probability theory, a interacting particle system is a stochastic process $(X(t))_{t\in \mathbb {R} ^{+}}$ on some Cartesian product $S=\times _{g\in G}S_{g}$ of discrete spaces $S_{g}$ (where $g\in G$ with $G$ a graph. It corresponds elementary stochastic processes $(X_{g}(t))_{t\in \mathbb {R} ^{+}}$ whose time evolution takes into account the evolution of some of the others elementary stochastic processes $(X_{g'}(t))_{t\in \mathbb {R} ^{+}}$ according to some interaction. Usually, time is continuous and the process is a Markov one. For discrete-time it is related to Markov chains and stochastic cellular automata.

Among the main examples are the voter model, the contact process, the asymmetric simple exclusion process (ASEP).

References

• Liggett, Thomas M. (1997). "Stochastic Models of Interacting Systems". The Annals of Probability. 25 (1). Institute of Mathematical Statistics: 1–29. doi:10.2307/2959527. ISSN 0091-1798.
• Liggett, Thomas M. (1985). Interacting Particle Systems. New York: Springer Verlag. ISBN 0-387-96069-4.