quenched speed is an observation that the main contribution to ${\mathbb{E}}[
E^\omega(\tau_1\,|\,\tau_1<\infty)]$ comes from paths which hit $1$ before entering $(-\infty,-a_1]$ where $-a_1=\max\{x\le 0:
V(x,\cdot)=M\}$ at some positive time. Our first step is to compute the main term.

\[escape\] For every $a_1\in{\mathbb{N}}\cup\{0\}$ and $p\in(0,1]$$${\mathbb{E}}[
  E^\omega(\tau_1\,|\,\tau_1<\tau_{-a_1})] = \frac{2-p+2p^2}{3p}.$$

Notice that our definition of $\tau_0$ (see (\[taux\])) implies $E^\omega(\tau_1\,|\,\tau_1<\tau_0)=1$. By Proposition \[basic1\] we have $$\begin{aligned}
    {\mathbb{E}}[E^\omega(\tau_1\,|\,\tau_1<\tau_{-a_1})]=
    p+p\sum_{a=1}^\infty
    \frac{2a_1+1}{3}\,(1-p)^{a_1}=
    p+\frac{1-p}{3}\left(\frac{2}{p}+1\right)= \frac{2-p+2p^2}{3p}.
  \end{aligned}$$

Given $\omega$, let $-a_1>-a_2>\dots$ be the occupied sites in $(-\infty,0]$, $I_j=[-a_{j+1},-a_j)$, $j\in{\mathbb{N}}$, and $I=[-a_1,1)$. Then $$\begin{gathered}
\label{dec1}
  {\mathbb{E}}[ E^\omega(\tau_1\,|\,\tau_1<\infty)]={\mathbb{E}}[
  E^\omega(\tau_1{\mathbbm{1}}_{\{\tau_1<\tau_{-a_1}\}}\,|\,\tau_1<\infty)]
  \\ +{\mathbb{E}}[
  E^\omega({\mathbbm{1}}_{\{\tau_1>\tau_{-a_1}\}}\sum_{n=0}^{\tau_1-1}{\mathbbm{1}}_{\{S_n\in
    I\}}\,|\,\tau_1<\infty)]+ \sum_{j=1}^\infty{\mathbb{E}}[
  E^\omega({\mathbbm{1}}_{\{\tau_1>\tau_{-a_1}\}}\sum_{n=0}^{\tau_1-1}{\mathbbm{1}}_{\{S_n\in
    I_j\}}\,|\,\tau_1<\infty)].\end{gathered}$$ Observe also that the first term in the right hand side of (\[dec1\]) equals $$\begin{gathered}
\label{dec2}
  {\mathbb{E}}[ E^\omega(\tau_1\,|\,\tau_1<\tau_{-a_1})
  P^{\omega}(\tau_1<\tau_{-a_1}\,|\,\tau_1<\infty)]=
  \\{\mathbb{E}}[E^\omega(\tau_1\,|\,\tau_1<\tau_{-a_1})]-
  {\mathbb{E}}[E^\omega(\tau_1\,|\,\tau_1<\tau_{-a_1})P^{\omega}(\tau_1>\tau_{-a_1}\,|\,\tau_1<\infty)].\end{gathered}$$ We shall need the following three elementary lemmas.

\[elem1\] For $a_1\in{\mathbb{N}}\cup\{0\}$ $$P^{\omega}(\tau_1>\tau_{-a_1}\,|\,\tau_1<\infty)\le
    e^{-M}\wedge ((2a_1(e^M-1)+1)(1+a_1))^{-1}.$$

\[elem2\] There is a constant $C_1$ such that $$E^\omega({\mathbbm{1}}_{\{\tau_1>\tau_{-a_1}\}}\sum_{n=0}^{\tau_1-1}{\mathbbm{1}}_{\{S_n\in
    I\}}\,|\,\tau_1<\infty)\le C_1(e^M-1)^{-1}.$$

\[elem3\] There is a constant $C_2$ such that for every $j\in{\mathbb{N}}$ $$E^\omega({\mathbbm{1}}_{\{\tau_1>\tau_{-a_1}\}}\sum_{n=0}^{\tau_1-1}{\mathbbm{1}}_{\{S_n\in
    I_j\}}\,|\,\tau_1<\infty)\le \frac{C_2e^{-Mj}} 
  {(1-e^{-M})^2}\frac{|I_j|}{|I|^2},$$ where $|A|$ is the Lebesgue measure of the set $A$.

Let us assume these facts (see Appendix for proofs) and derive Theorem \[vque\].

The right hand side of the inequality in Lemma \[elem1\] does not exceed $e^{-M}/(1+a_1)$. Therefore, $$\begin{gathered}
    {\mathbb{E}}[E^\omega(\tau_1\,|\,\tau_1<\tau_{-a_1}) 
P^{\omega}(\tau_1>\tau_{-a_1}\,|\,\tau_1<\infty)]\le\\
    e^{-M}+pe^{-M}\sum_{a_1=1}^\infty
    (1-p)^{a_1}\frac{2a_1+1}{3(a_1+1)}<
    