Mathematics – Probability
Scientific paper
2012-03-07
Mathematics
Probability
11 pages, 1 figure
Scientific paper
We study a trapping problem on Z^d with mobile traps, where a particle starting from the origin is killed when it first encounters a trap. The traps are distributed as a Poisson point process on Z^d, each moving independently as a simple symmetric random walk with i.i.d. holding times, where heavy-tailed holding times give rise to sub-diffusive trap motion. It is accepted in the physics literature that among all deterministic trajectories the particle may follow up to time t, the constant trajectory maximizes the survival probability. This is known as the Pascal principle. Previously, the Pascal principle has been verified rigorously for exponential holding times. In this note, we extend it to holding times with a general continuous distribution. As a byproduct, we find that the expected number of sites visited by an n-step simple symmetric random walk can only increase if deterministic jumps are inserted in the path of the walk. We conjecture that this holds for all symmetric random walks on Z^d.
Chen Lung-Chi
Sun Rongfeng
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