The survival of branching annihilating random walk. (English) Zbl 0537.60099
Branching annihilating random walk is an interacting particle system on \({\mathbb{Z}}\). As time evolves, particles execute random walks and branch, and disappear when they meet other particles. It is shown that starting from a finite number of particles, the system will survive with positive probability if the random walk rate is low enough relative to the branching rate, but will die out with probability one if the random walk rate is high. Since the branching annihilating random walk is non- attractive, standard techniques usually employed for interacting particle systems are not applicable. Instead, a modification of a contour argument by L. Gray and D. Griffeath [Ann. Probab. 10, 67-85 (1982; Zbl 0483.60090)] is used.
MSC:
60K35 | Interacting random processes; statistical mechanics type models; percolation theory |
60G50 | Sums of independent random variables; random walks |
Citations:
Zbl 0483.60090References:
[1] | Bramson, M. D.; Gray, L. F., A note on the survival of the long-range contact process, Ann. Probability, 9, 885-890 (1981) · Zbl 0467.60090 |
[2] | Gray, L. F.; Griffeath, D., A stability criterion for attractive nearest neighbor spin systems on Z, Ann. Probability, 10, 67-85 (1982) · Zbl 0483.60090 |
[3] | Griffeath, D., Additive and Cancellative Interacting Particle Systems, Lecture Notes in Mathematics 724 (1979), Berlin-Heidelberg-New York: Springer, Berlin-Heidelberg-New York · Zbl 0412.60095 |
[4] | Liggett, T. M., The stochastic evolution of infinite systems of interacting particles, Lecture Notes in Mathematics 598, 187-248 (1977), Berlin-Heidelberg-New York: Springer, Berlin-Heidelberg-New York · Zbl 0363.60109 |
[5] | Stroock, D.: Lectures on Infinite Interacting Particle Systems. Kyoto University Lectures in Mathematics 11, 1978 · Zbl 0407.60097 |
[6] | Ulam, S.M.: The role of mathematical abstraction in the physical sciences. MAA invited address at the 59th Summer Meeting, Duluth, 1979 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.