Electronic Journal of Probability

A Note on Limiting Behaviour of Disastrous Environment Exponents

Thomas Mountford

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We consider a random walk on the $d$-dimensional lattice and investigate the asymptotic probability of the walk avoiding a "disaster" (points put down according to a regular Poisson process on space-time). We show that, given the Poisson process points, almost surely, the chance of surviving to time $t$ is like $e^{-\alpha \log (\frac1k) t } $, as $t$ tends to infinity if $k$, the jump rate of the random walk, is small.

Article information

Electron. J. Probab., Volume 6 (2001), paper no. 1, 10 pp.

Accepted: 5 January 2001
First available in Project Euclid: 19 April 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Random walk disaster point Poisson process

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Mountford, Thomas. A Note on Limiting Behaviour of Disastrous Environment Exponents. Electron. J. Probab. 6 (2001), paper no. 1, 10 pp. doi:10.1214/EJP.v6-74. https://projecteuclid.org/euclid.ejp/1461097631

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  • Durrett, R., Lecture Notes on Particle Systems and Percolation. Wadsworth, Pacific Grove. (1988).
  • Durrett, R., Oriented Percolation in two dimensions. The Annals of Probabability. 12, 999-1040, (1984),
  • Durrett, R., Ten Lectures on Particle Systems. École d'Été de St. Flour. XXIII Springer, New York, Berlin, (1993).
  • Bezuidenhout, C. and Grimmett, G., The critical contact process dies out. The Annals of Probability. 18, 1990, 1462-1482.
  • Liggett, T.M. Interacting Particle Systems. Springer, Berlin, New York, (1985).
  • Shiga, T., Exponential decay rate of survival probability in a disastrous random environment. Theory of Probability and Related Fields. 108, 1997, 417-439.