## 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)

### Keywords:

random walk; disaster point; Poisson process
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