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

Abstract

We review recent studies demonstrating a nonuniversal (continuously variable) survival exponent for history-dependent random walks, and analyze a new example, the hard movable partial reflector. These processes serve as a simplified models of infection in a medium with a history-dependent susceptibility, and for spreading in systems with an infinite number of absorbing configurations. The memory may take the form of a history-dependent step length, or be the result of a partial reflector whose position marks the maximum distance the walker has ventured from the origin. In each case, a process with memory is rendered Markovian by a suitable expansion of the state space. Asymptotic analysis of the probability generating function shows that, for large t, the survival probability decays as S(t) \sim t^{-delta}, where \delta varies with the parameters of the model. We report new results for a hard partial reflector, i.e., one that moves forward only when the walker does. When the walker tries to jump to the site R occupied by the reflector, it is reflected back with probability r, and stays at R with probability 1-r; only in the latter case does the reflector move (R \to R+1). For this model, delta = 1/2(1-r), and becomes arbitrarily large as r approaches 1. This prediction is confirmed via iteration of the transfer matrix, which also reveals slowly-decaying corrections to scaling.