Crossing speeds of random walks among "sparse" or "spiky" Bernoulli potentials on integers

TL;DR

This paper investigates how the asymptotic speed of a one-dimensional random walk among Bernoulli obstacles varies in sparse and spiky obstacle regimes, revealing significant differences between quenched and annealed measures.

Contribution

It provides a detailed analysis of the asymptotic speed behavior in sparse and spiky obstacle regimes, highlighting the contrast between quenched and annealed settings.

Findings

As p approaches 0 with fixed M, the speed decreases significantly.

As M approaches infinity with fixed p, the speed behavior differs markedly from the sparse case.

Quenched and annealed measures exhibit notably different asymptotic behaviors.

Abstract

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.