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

TL;DR

This paper analyzes a history-dependent random walk with variable step lengths, deriving its asymptotic survival probability decay rate and confirming the results through numerical methods.

Contribution

It introduces a simplified model of spreading with history-dependent steps and derives the decay exponent for survival probability asymptotically.

Findings

Survival probability decays as t^{-v/2n} for large t.

Asymptotic analysis matches numerical transition matrix results.

Provides insights into systems with infinite absorbing configurations.

Abstract

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.