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A note on limiting behaviour of disastrous environment exponents. (English) Zbl 0976.60093

Summary: 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 \(e^{-\alpha \log (1/k) t } \), as \(t\) tends to infinity if \(k\), the jump rate of the random walk, is small.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60G50 Sums of independent random variables; random walks
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
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