  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.
