  We consider a random walk among i.i.d. obstacles on the one dimensional
integer lattice under the condition that the walk starts from the origin and
reaches a remote location y. The obstacles are represented by a killing
potential, which takes value M>0 with probability p and value 0 with
probability (1-p), 0<p<1, independently at each site of the lattice. We
consider the walk under both quenched and annealed measures. It is known that
under either measure the crossing time from 0 to y of such walk, tau(y), grows
linearly in y. More precisely, the expectation of tau(y)/y converges to a limit
as y approaches infinity. The reciprocal of this limit is called the asymptotic
speed of the conditioned walk. We study the behavior of the asymptotic speed in
two regimes: (1) as p goes to 0 for M fixed ("sparse"), and (2) as M goes to
infinity for p fixed ("spiky"). We observe and quantify a dramatic difference
between the quenched and annealed settings.
