A Note on Limiting Behaviour of Disastrous Environment Exponents

Thomas S. Mountford (University of California, Los Angeles)


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.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1-10

Publication Date: January 5, 2001

DOI: 10.1214/EJP.v6-74


  1. Durrett, R. , Lecture Notes on Particle Systems and Percolation. Wadsworth, Pacific Grove. (1988). Math. Review number MR89h:60157
  2. Durrett, R., Oriented Percolation in two dimensions. The Annals of Probabability. 12, 999-1040, (1984), Math. Review number MR86g:60117
  3. Durrett, R., Ten Lectures on Particle Systems. École d'Été de St. Flour. XXIII Springer, New York, Berlin, (1993). Math. Review number MR96k:60004
  4. Bezuidenhout, C. and Grimmett, G., The critical contact process dies out. The Annals of Probability. 18, 1990, 1462-1482. Math. Review number MR91k:60111
  5. Liggett, T.M. Interacting Particle Systems. Springer, Berlin, New York, (1985).
  6. Shiga, T. , Exponential decay rate of survival probability in a disastrous random environment. Theory of Probability and Related Fields. 108, 1997, 417-439. Math. Review number MR98f:60212

Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.