Winning quick and dirty: the greedy random walk

Details Winning quick and dirty: the greedy random walk Winning quick and dirty: the greedy random walk

2004-09-03

arxiv.org/abs/cond-mat/0409060v1

Journal of Physics A 37, 11321-11331 (2004)

Physics

Condensed Matter

Statistical Mechanics

7 pages, 6 figures, 2-column revtext4 format

Scientific paper

10.1088/0305-4470/37/47/002

As a strategy to complete games quickly, we investigate one-dimensional random walks where the step length increases deterministically upon each return to the origin. When the step length after the kth return equals k, the displacement of the walk x grows linearly in time. Asymptotically, the probability distribution of displacements is a purely exponentially decaying function of |x|/t. The probability E(t,L) for the walk to escape a bounded domain of size L at time t decays algebraically in the long time limit, E(t,L) ~ L/t^2. Consequently, the mean escape time ~ L ln L, while ~ L^{2n-1} for n>1. Corresponding results are derived when the step length after the kth return scales as k^alpha$ for alpha>0.

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