Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-25T08:09:00.351Z Has data issue: false hasContentIssue false
  • English
  • Français

Phase transitions and duality properties of a successional model

Part of: Special processes

Published online by Cambridge University Press:  01 July 2016

Nicolas Lanchier
Nicolas Lanchier*
Affiliation:
Université de Rouen
*
∗ Postal address: Laboratoire de Mathématiques Raphaël Salem, UMR 6085, Université de Rouen, 76128 Mont Saint Aignan, France. Email address: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The first purpose of this article is to study the phase transitions of a new interacting particle system. We consider two types of particle, each of which gives birth to particles of the same type as the parent. Particles of the second type can die, whereas those of the first type mutate into the second type. We prove that the three possible outcomes of the process, that is, extinction, survival of the type-2s, or coexistence, may each occur, depending on the selected parameters. Our second, and main, objective, however, is to investigate the duality properties of the process; the corresponding dual process exhibits a structure somewhat different to that of well-known particle systems.

Keywords

Interacting particle systemcompetition modelphase transitionmultitype contact processforest fire modelduality

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 37 , Issue 1 , March 2005 , pp. 265 - 278
Copyright
Copyright © Applied Probability Trust 2005 

References

Durrett, R. (1984). Oriented percolation in two dimensions. Ann. Prob. 12, 999–1040.CrossRefGoogle Scholar
Durrett, R. (1988). Lecture Notes on Particle Systems and Percolation. Wadsworth and Brooks, Pacific Grove, CA.Google Scholar
Durrett, R. (1995). Ten lectures on particle systems. In Lectures on Probability Theory (Saint-Flour, 1993; Lecture Notes Math. 1608), Springer, Berlin, pp. 97–201.Google Scholar
Durrett, R. and Neuhauser, C. (1991). Epidemics with recovery in d = 2. Ann. Appl. Prob. 1, 189–206.Google Scholar
Durrett, R. and Neuhauser, C. (1997). Coexistence results for some competition models. Ann. Appl. Prob. 7, 10–45.Google Scholar
Harris, T. E. (1972). Nearest-neighbor Markov interaction processes on multidimensional lattices. Adv. Math. 9, 66–89.Google Scholar
Harris, T. E. (1976). On a class of set-valued Markov processes. Ann. Prob. 4, 175–194.Google Scholar
Kuczek, T. (1989). The central limit theorem for the right edge of supercritical oriented percolation. Ann. Prob. 17, 1322–1332.Google Scholar
Liggett, T. M. (1999). Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Springer, Berlin.Google Scholar
Neuhauser, C. (1992). Ergodic theorems for the multitype contact process. Prob. Theory Relat. Fields 91, 467–506.Google Scholar