Oriented percolation in two dimensions
This paper is a self-contained survey of most of the results known about oriented percolation. The table of contents below should give an idea of the topics which will be covered. 1. A few words of explanation. 2. Description of the model. 3. A useful characterization of \(p_ c\). 4. A stationary measure for the edge process. 5. Recurrence properties of \(\bar r_ n\), lower bounds on \(P(\Omega_{\infty})\). 6. Lower bounds on \(p_ c\). 7. Exponential estimates for \(p<p_ c\). 8. Time reversal duality, first results for \(p>p_ c\). 9. A construction for studying \(p>p_ c\) (and showing \(\alpha (p_ c)=0)\). 10. \(P(\tau^ A<\infty)\leq C \exp (-\gamma | A|)\). 11. Large deviation results for \(\bar r_ n\). 12. Bounds on \(P(n\leq \tau^ A<\infty)\) and \(P(n\leq | C_ 0| <\infty)\). 13. Correlation inequalities for \(\xi^ Z_{\infty}\), limit laws for \(| \xi^ o_ n|\). 14. Infinite differentiability of \(P(\Omega_{\infty})\) for \(p>p_ c.\) We have written this article with the hope of acquainting people and with what is known and more important and with what is not known. With the last purpose in mind, this paper is aimed at two audiences: (1) people who are experts on the subject and (2) people who are not but would like to learn something about it or the broader topic: interacting particle systems.
scientific article; zbMATH DE number 3905614