Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-17T19:06:20.010Z Has data issue: false hasContentIssue false
  • English
  • Français

The annihilating process

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

Martin O'Hely  and
Aidan Sudbury
Martin O'Hely*
Affiliation:
University of Oregon
Aidan Sudbury*
Affiliation:
Monash University
*
∗ Postal address: Department of Biology, University of Oregon, Engene, OR 97403, USA.
∗∗ Postal address: Department of Mathematics and Statistics, Monash University, PO Box 28M, Victoria 3800, Australia. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

An annihilating process is an interacting particle system in which the only interaction is that a particle may kill a neighbouring particle. Since there is no birth and no movement, once a particle has no neighbours its site remains occupied for ever. It is shown that with initial configuration ℤ the distribution of particles at all times is a renewal process and that the probability that a site remains occupied for all time tends to 1/e. Time-dependent behaviour is also calculated for the tree 𝕋r.

Keywords

infinite particle systemannihilation

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 38 , Issue 1 , March 2001 , pp. 223 - 231
Copyright
Copyright © by the Applied Probability Trust 2001 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Arratia, R. (1981). Limiting point processes for rescaling of coalescing and annihilating random walks on Z d . Ann. Prob. 9, 909–936.CrossRefGoogle Scholar
Bramson, M., and Gray, L. (1985). The survival of the branching annihilating random walk. Z. Wahrscheinlichskeitsth. 68, 447–460.CrossRefGoogle Scholar
Bramson, M., and Griffeath, D. (1980). Asymptotics for interacting particle systems on Z d . Z. Wahrscheinlichskeitsth. 53, 183–196.CrossRefGoogle Scholar
Daley, D. J., Mallows, C. L., and Shepp, L. A. (2000). A one-dimensional Poisson growth model with non-overlapping intervals. Stoch. Proc. Appl. 90, 223–241.CrossRefGoogle Scholar
Neuhauser, C., and Sudbury, A. W. (1993). The biased annihilating branching process. Adv. Appl. Prob. 25, 24–38.CrossRefGoogle Scholar