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

Abstract

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.

Scaling behavior for random walks with memory of the largest distance from the origin

“…In Refs. [20,21] and in Ref. [22], as in the model presented in this paper, the behavior of the walker is modified only when it is at the maximum distance from the origin and Markovianity is recovered only when the phase space is properly enlarged.…”

“…Random walks with memory have been also employed to model the spreading of an infection in a medium with a history-dependent susceptibility [20,21]; the focus, in this case, is the time scaling of the survival probability (a trap is collocated somewhere) and not the scaling of diffusion. Moreover, random walks with memory have been used in finance as, for example, in Ref.…”

Continuously variable spreading exponents in the absorbing Nagel–Schreckenberg model

“…But this is not the only possible consistent set of spreading exponents. Here, surprisingly, I find that δ and η vary continuously, in a situation reminiscent of one-dimensional random walks with an absorbing boundary at the origin and a moveable reflector [17] or history-dependent step length [18], or CDP subject to moveable reflectors [19]. In these cases the survival exponent δ varies continuously with a parameter.…”

Variable survival exponents in history-dependent random walks: hard movable reflector

“…[18] were applied to a one-dimensional random walk with memory of a different form: if the target site x lies in the region that has been visited before (that is, if x itself has been visited, or lies between two sites that have been visited), then the step length is v; otherwise the step length is n. In this case one finds δ = v/2n [20]. Thus δ can take any rational value between zero and infinity.…”

“…[18] and [20]. After formulating the problem, we enlarge the state space so that the process becomes Markovian in the expanded representation [21].…”