  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.
