Asymptotic analysis of a random walk with a history-dependent step length

Details Asymptotic analysis of a random walk with a history-dependent step length Asymptotic analysis of a random walk with a history-dependent step length

2002-06-14

arxiv.org/abs/cond-mat/0206281v2

Phys. Rev. E 66, 051102 (2002)

Physics

Condensed Matter

Statistical Mechanics

15 pages, 2 figures

Scientific paper

10.1103/PhysRevE.66.051102

We study an unbiased, discrete time random walk on the nonnegative integers, with the origin absorbing. The process has a history-dependent step length: the walker takes steps of length v while in a region which has been visited before, and steps of length n when entering a region that has never been visited. The process provides a simplified model of spreading in systems with an infinite number of absorbing configurations. Asymptotic analysis of the probability generating function shows that, for large t, the survival probability decays as S(t) \sim t^{-delta}, with delta = v/2n. Our expression for the decay exponent is in agreement with results obtained via numerical iteration of the transition matrix.

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