Crossing Speeds of Random Walks Among 'Sparse' or 'Spiky' Bernoulli Potentials on Integers.

We consider a random walk among i.i.d. obstacles on $\mathbb {Z}$ 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 $\mathbb {Z}$. 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, τ, grows linearly in y. More precisely, the expectation of τ/ y converges to a limit as y→∞. 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→0 for M fixed ('sparse'), and (2) as M→∞ for p fixed ('spiky'). We observe and quantify a dramatic difference between the quenched and annealed settings. [ABSTRACT FROM AUTHOR]