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

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(t) approximately t(-delta), with delta=v/2n. Our expression for the decay exponent is in agreement with the results obtained via numerical iteration of the transition matrix.