We study an unbiased, discrete-time random walk on the nonnegative integers, with the origin absorbing, and a history-dependent step length. Letting y denote the maximum distance the walker has ever been from the origin, steps that do not change y have length $v,$ while those that increase y (taking the walker to a site that has never been visited) have length n. The process serves as 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 $S\left(t\right)\mathrm{}\sim t^{-\delta },$ with $\delta =v/2n.\mathrm{}$ Our expression for the decay exponent is in agreement with the results obtained via numerical iteration of the transition matrix.

• Received 14 June 2002

DOI:https://doi.org/10.1103/PhysRevE.66.051102