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The annihilating process

  • Martin O'Hely (a1) and Aidan Sudbury (a2)

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 𝕋 r .

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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: aidan.sudbury@sci.monash.edu.au

References

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Arratia, R. (1981). Limiting point processes for rescaling of coalescing and annihilating random walks on Z d . Ann. Prob. 9, 909936.
Bramson, M., and Gray, L. (1985). The survival of the branching annihilating random walk. Z. Wahrscheinlichskeitsth. 68, 447460.
Bramson, M., and Griffeath, D. (1980). Asymptotics for interacting particle systems on Z d . Z. Wahrscheinlichskeitsth. 53, 183196.
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, 223241.
Neuhauser, C., and Sudbury, A. W. (1993). The biased annihilating branching process. Adv. Appl. Prob. 25, 2438.

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The annihilating process

  • Martin O'Hely (a1) and Aidan Sudbury (a2)

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