Oriented percolation in two dimensions
type: Item
instance of: scholarly article
title: Oriented percolation in two dimensions
zbMATH Open document ID: 0567.60095
published in: The Annals of Probability
publication date: 1984-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 82B43
Mathematics Subject Classification ID: 60F10
zbMATH DE Number: 3905614
zbMATH Keywords: survey
zbMATH Keywords: oriented percolation
zbMATH Keywords: Recurrence properties
zbMATH Keywords: Time reversal duality
zbMATH Keywords: Correlation inequalities
zbMATH Keywords: interacting particle systems
author: Rick Durrett
MaRDI profile type: Publication
Wikidata QID: Q61631798
full work available at URL: https://doi.org/10.1214/aop/1176993140
OpenAlex ID: W1967121312
DOI: 10.1214/AOP/1176993140
description: scientific article; zbMATH DE number 3905614