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## Brazilian Journal of Physics

*Print version* ISSN 0103-9733*On-line version* ISSN 1678-4448

#### Abstract

DICKMAN, Ronald; ARAUJO JR., Francisco Fontenele and BEN-AVRAHAM, Daniel. **Variable survival exponents in history-dependent random walks**: **hard movable reflector**.* Braz. J. Phys.* [online]. 2003, vol.33, n.3, pp.450-457.
ISSN 0103-9733. http://dx.doi.org/10.1590/S0103-97332003000300006.

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 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 historydependent 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*) ~ *t *^{-}^{d}, where d 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 ® R+1). For this model, d = 1/2(1 - *r*), and becomes arbitrarily large as *r* approaches 1. This prediction is confirmed via iteration of the transition matrix, which also reveals slowly-decaying corrections to scaling.