i.i.d. random variables satisfying (\[pot\]) and $v^{{\mathrm{que}}}_{V}$ and $v^{{\mathrm{ann}}}_{V}$ be the quenched and annealed speeds as defined in (\[speeds\]). Using our methods it should be not difficult to show that as $\gamma\downarrow 0$ $$\label{op1}
  v^{{\mathrm{que}}}_{\gamma V},\ v^{{\mathrm{ann}}}_{\gamma V}\sim
\sqrt{2\gamma{\mathbb{E}}(V)}.$$ This result would complement the results of this paper. The relation (\[op1\]) is suggested by the following two facts.

\(i) The speeds can be equivalently defined as follows (Proposition \[basic1\] below and [@KM12 Theorem 1.2]): $$\label{sle}
  \frac{1}{v^{{\mathrm{que}}}_V}=\frac{d}{d\lambda}\alpha_{\lambda+V}(1)\Big|_{\lambda=0+},
  \quad
  \frac{1}{v^{{\mathrm{ann}}}_V}=\frac{d}{d\lambda}\beta_{\lambda+V}(1)\Big|_{\lambda=0+},$$ where for each $\lambda\ge 0$ the non-random quantities $$\begin{aligned}
  \alpha_{\lambda+V}(1)&:=-\lim_{y\to\infty}\frac1y \log
  E\left({\mathbbm{1}}_{\{\tau_y<\infty\}}e^{-\sum_{n=0}^{\tau_y-1}(\lambda+V(S_n,\omega))}\right)
  \quad
  (\text{${\mathbb{P}}$-a.s.});\label{que}\\
  \beta_{\lambda+V}(1)&:=-\lim_{y\to\infty}\frac1y
  \log{\mathbb{E}}\left[E\left({\mathbbm{1}}_{\{\tau_y<\infty\}}e^{-\sum_{n=0}^{\tau_y-1}
        (\lambda+V(S_n,\omega))}\right)\right]\label{ann}\end{aligned}$$ are the quenched and annealed (respectively) Lyapunov exponents of a random walk in the potential $\lambda+V$. Under our assumptions, both limits are positive and finite. For more details about the existence and properties of Lyapunov exponents see [@Ze98], [@Fl07], and [@Mou12].

\(ii) It was shown in [@KMZ11] (see also [@Wa01], [@Wa02]) that $\alpha_{\gamma V}(1),\ \beta_{\gamma V}(1)\sim
\sqrt{2\gamma{\mathbb{E}}(V)}$ as $\gamma\downarrow 0$. Since $$\alpha_{\gamma(\lambda+V)}(1)\sim\sqrt{2\gamma(\lambda+{\mathbb{E}}(V))} 
\quad\text{as $\gamma\downarrow 0$ \quad and}\quad \frac{d}{d(\gamma
  \lambda)}=\frac1\gamma \frac{d}{d\lambda},$$ a formal differentiation of $\sqrt{2\gamma(\lambda+{\mathbb{E}}(V))}$ with respect to $\lambda$ and division by $\gamma$ lead to the conjecture (\[op1\]). We remark that in the case of a constant potential (without loss of generality we set $V\equiv 