Measure $P$, and the corresponding expectation $E$, will always refer to the simple symmetric random walk on ${\mathbb{Z}}$ starting from $0$.

Next, fix $y>0$ and consider the conditional measure on nearest neighbor paths starting from $0$, which is defined by $$\label{Qq}
  Q_y^{\omega}(\,\cdot\,):={P}^{\omega}(\,\cdot\,|\,\tau_y<\infty).$$ Measure $Q_y^{\omega}$ is called the quenched path measure. Notice that it can be equivalently defined as follows: $$\begin{aligned}
  \label{qpm}
  Q^{\omega}_y(A):&=(Z^{\omega}_y)^{-1}E\left({\mathbbm{1}}_{\{\tau_y<\infty\}}
    {\mathbbm{1}}_A\,e^{-\sum_{n=0}^{\tau_y-1}V(S_n,\omega)}\right),\ \text{where
  }
  \\\label{qnf} 
Z^{\omega}_y&:={P}^\omega(\tau_y<\infty)=E\left({\mathbbm{1}}_{\{\tau_y<\infty\}}
    e^{-\sum_{n=0}^{\tau_y-1}V(S_n,\omega)}\right).\end{aligned}$$

Finally, we define the annealed measure $Q_y$, which is a measure on the product space of $\Omega$ and the nearest-neighbor paths starting from $0$. We let $$\label{am}
    Q_y(B):=\dfrac{{\mathbb{E}}({P}^\omega 
(B\cap\{\tau_y<\infty\}))}{{\mathbb{E}}({P}^\omega(\tau_y<\infty))}.$$ Whenever the starting point of a process is different from $0$, we shall indicate it with a superscript, for example, $Q^{\omega,x}_y$ will denote a quenched path measure of a killed random walk, which starts at $x$ and is conditioned to hit $y$, $y>x$.

The quenched and annealed asymptotic speeds, $v^{{\mathrm{que}}}_V$ and $v^{{\mathrm{ann}}}_V$, are the deterministic quantities, defined by $$\label{speeds}
  \frac{1}{v^{{\mathrm{que}}}_V}:=\lim_{y\to\infty}\frac{E_{Q^{\omega}_y}(\tau_y)}{y}\quad
  ({\mathbb{P}}\text{-a.s.});\quad
  \frac{1}{v^{{\mathrm{ann}}}_V}:=\lim_{y\to\infty}\frac{E_{Q_y}(\tau_y)}{y}.$$ By Proposition \[basic1\] below and [@KM12 Theorem 1.2] respectively these limits exist and are finite.

In this paper we consider i.i.d.  Bernoulli potentials $V_{p,M}(x,\cdot)$, $x\in{\mathbb{Z}}$, $p\in(0,1]$, $M>0$, $$\label{bp}
  {\mathbb{P}}(V_{p,M}(x,\cdot)=M)=1-{\mathbb{P}}(V_{p,M}(x,\cdot)=0)=p,$$ and study the behavior of the corresponding quenched and annealed asymptotic speeds 