is $p_0=p_0({\varepsilon})$ such that for each $p<p_0$ there is $y_0=y_0(p)$ such that for all $y>y_0$

-   $Q_{0,y}(\Omega^1_y)\ge 1/2$ and

-   for every $\omega\in\Omega^1_y$ $$E_{Q_{0,y}^\omega}(\tau_y)\ge
        C_1ye^{(1-{\varepsilon})K(p,M)},$$

where $C_1$ does not depend on $y,\omega,p$. Then $$\begin{gathered}
\frac{E_{Q_{0,y}}(\tau_y)}{y}\ge
(yZ_{0,y})^{-1}{\mathbb{E}}(E_{Q^\omega_{0,y}}(\tau_y)Z^\omega_{0,y};
\Omega^1_y) \ge C_1e^{(1-{\varepsilon})K(p,M)}\,
Q_{0,y}(\Omega^1_y)\ge \frac{C_1}{2}\,e^{(1-{\varepsilon})K(p,M)},\end{gathered}$$ and, hence, $$\label{lb}
  -\liminf_{p\to 0}p\log v^{ann}_{p,M}\ge
(1-{\varepsilon})\lim_{p\to 0}pK(p,M)=2(1-{\varepsilon})(e^M-1).$$ For an upper bound we shall consider environments $\Omega^2_y=\Omega^2_y({\varepsilon},p,M)$ for which the following holds: for every ${\varepsilon}>0$ there is $p_0=p_0({\varepsilon})$ such that for each $p<p_0$

-   $\lim\limits_{y\to\infty}y\,Q_{0,y}(\Omega\setminus
      \Omega^2_y)=0$ and

-   there is $y_0=y_0(p)$ such that for all $y>y_0$, $\omega\in \Omega^2_y$ $$E_{Q_{0,y}^\omega}(\tau_y)\le C_2y
      e^{(1+{\varepsilon})K(p,M)},$$

where $C_2$ does not depend on $y,\omega,p$. Then $$\begin{gathered}
\frac{E_{Q_{0,y}}(\tau_y)}{y}\le
(yZ_{0,y})^{-1}\left({\mathbb{E}}(E_{Q^\omega_{0,y}}(\tau_y)Z^\omega_{0,y};
\Omega^2_y)
+{\mathbb{E}}(E_{Q^\omega_{0,y}}(\tau_y)Z^\omega_{0,y};
\Omega\setminus\Omega^2_y)\right)\\ \le
C_2e^{(1+{\varepsilon})K(p,M)}+(yZ_{0,y})^{-1}{\mathbb{E}}(E_{Q^\omega_{0,y}}(\tau_y)Z^\omega_{0,y};
\Omega\setminus\Omega^2_y).\end{gathered}$$ By Lemma \[A\] (see Appendix), $E_{Q^\omega_{0,y}}(\tau_y)\le 3y^2$. Combining this with (U1) we get $$\label{ub}
  -\limsup_{p\to 0}p\log v^{ann}_{p,M}\le
(1+{\varepsilon})\lim_{p\to 0}pK(p,M)=2(1+{\varepsilon})(e^M-1).$$ Since ${\varepsilon}$ is arbitrary, relations (\[lb\]) and (\[ub\]) imply (\[ap\]). Our task will be to construct $\Omega^i_y$, $i=1,2$, with the desired properties. The starting point for obtaining (L2) and (U2) is Lemma \[U\] which gives bounds on $E_{Q^\omega_{0,y}}(\tau_y)$ in terms of gaps between obstacles. The construction of $\Omega^i_y$, $i=1,2$, will be carried out in Subsection \[last\] after we obtain information about a typical environment under the annealed measure $Q_{0,y}$. The latter is 