---
abstract: '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 $1-p$, $0<p\le 1$, 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, $\tau_y$, grows linearly in $y$. More precisely, the expectation of $\tau_y/y$ converges to a limit as $y\to\infty$. 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 $p\to 0$ for $M$ fixed (“sparse”), and (2) as $M\to \infty$ for $p$ fixed (“spiky”). We observe and quantify a dramatic difference between the quenched and annealed settings.'
address: 'Department of Mathematics, Baruch College, One Bernard Baruch Way, New York, NY 10010'
author:
- Elena Kosygina
title: 'Crossing speeds of random walks among “sparse” or “spiky” Bernoulli potentials on integers'
---

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Introduction
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We shall start with a model description 