Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-05T15:10:42.872Z Has data issue: false hasContentIssue false

The survival of various interacting particle systems

Published online by Cambridge University Press:  01 July 2016

Aidan Sudbury*
Affiliation:
Monash University
*
* Postal address: Department of Mathematics, Monash University, Clayton, VIC 3168, Australia.
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.

Particles may be removed from a lattice by murder, coalescence, mutual annihilation and simple death. If the particle system is not to die out, the removed particles must be replaced by births. This letter shows that coalescence can be counteracted by arbitrarily small birth-rates and contrasts this with the situations for annihilation and pure death where there are critical phenomena. The problem is unresolved for murder.

Type
Letter to the Editor
Copyright
Copyright © Applied Probability Trust 1993 

References

Arratia, R. (1981) Limiting point processes for rescaling of coalescing and annihilating random walks on. Ann. Prob. 9, 909936.CrossRefGoogle Scholar
Bramson, M. and Gray, L. (1985) The survival of the branching annihilating random walk. Z. Wahrscheinlichkeitsth. 68, 447460.Google Scholar
Bramson, M. and Griffeath, D. (1980) Asymptotics for interacting particle systems on d. Z. Wahrscheinlichkeitsth. 53, 183196.Google Scholar
Clifford, P. and Sudbury, A. (1979) On the use of bounds in the statistical analysis of spatial processes. Biometrika 66, 495504.Google Scholar
Durrett, R. (1988) Lecture Notes on Particle Systems and Percolation. Wadsworth and Brooks, Pacific Grove, California.Google Scholar
Harris, T. E. (1974) Contact interactions on a lattice. Ann. Prob. 2, 969988.CrossRefGoogle Scholar
Holley, R. and Liggett, T. M. (1978) The survival of contact processes. Ann. Prob. 6, 198206.Google Scholar
Mountford, T. (1993) A coupling of finite particle systems. J. Appl. Prob. 30, 258262.Google Scholar
Neuhauser, C. and Sudbury, A. (1993) The biased annihilating branching process. Adv. Appl. Prob. 25, 2438.Google Scholar