e^{-M}+\frac{2pe^{-M}}{3}\sum_{a_1=1}^\infty (1-p)^{a_1}\le
    \frac{5}{3}\,e^{-M}.
  \end{gathered}$$ This immediately gives $$\begin{aligned}
    &\lim_{p\to
      0}p\,{\mathbb{E}}[E^\omega(\tau_1\,|\,\tau_1<\tau_{-a_1})
    P^{\omega}(\tau_1>\tau_{-a_1}\,|\,\tau_1<\infty)]=0;\\
    &\lim_{M\to
      \infty}{\mathbb{E}}[E^\omega(\tau_1\,|\,\tau_1<\tau_{-a_1}) 
    P^{\omega}(\tau_1>\tau_{-a_1}\,|\,\tau_1<\infty)]=0.
  \end{aligned}$$ Lemma \[elem2\] takes care of the second term in the right hand side of (\[dec1\]). By Lemma \[elem3\] and the independence of the values of the potential at distinct sites, the last term in (\[dec1\]) is bounded by (we defined the function $(1-p)^{-1}\ln(1/p)$ to be $1$ at $p=1$ by continuity) $$\begin{gathered}
  \frac{C_2}{(1-e^{-M})^2}\sum_{j=1}^\infty
  e^{-Mj}{\mathbb{E}}[(a_{j+1}-a_j)]{\mathbb{E}}[(a_1+1)^{-2}]\\\le\frac{C_2}
  {(1-e^{-M})^2}\sum_{j=1}^\infty e^{-Mj}\,\frac{\ln
    (1/p)}{1-p}=\frac{C_2e^{-M}\ln
   (1/p)}{(1-p)(1-e^{-M})^3}.\end{gathered}$$

This expression clearly vanishes as $M\to \infty$ locally uniformly in $p\in(0,1]$. After multiplication by $p$ it converges to $0$ as $p\to
0$ uniformly on every interval $[M_0,\infty)$. The only term left in the right hand side of (\[dec1\]) and (\[dec2\]) is the main term, ${\mathbb{E}}[ E^\omega(\tau_1\,|\,\tau_1<\tau_{-a_1})]$, which has the claimed asymptotics by Lemma \[escape\].

Annealed speed
==============

We start by introducing additional notation. Let $$Q_{0,y}(\cdot):=Z_{0,y}^{-1}{\mathbb{E}}[P^\omega (\ \cdot\
;\tau_y<\tau_0,\tau_y<\infty)],\quad
Z_{0,y}:={\mathbb{E}}[P^\omega(\tau_y<\tau_0,\tau_y<\infty)].$$ The corresponding quenched path measure $Q_{0,y}^\omega$ is given by $$Q_{0,y}^\omega(\cdot)=(Z^\omega_{0,y})^{-1} {P}^\omega(\ \cdot\
;\tau_y<\tau_0,\tau_y<\infty),\quad Z^\omega_{0,y}:=
{P}^\omega(\tau_y<\tau_0,\tau_y<\infty).$$

Heuristics and goals.
---------------------

Our first observation is that we can replace the measure $Q_y$ with $Q_{0,y}$ (see Proposition 3.1, (3.4), and the proof of (1.7) in [@KM12]). The key ingredient of the proof of Theorem \[vann\] is the study of environments under $Q_{0,y}$. 