1$) it is easy to compute the Lyapunov exponent explicitly: $\alpha_{\lambda+\gamma}(1)=\beta_{\lambda+\gamma}(1)=\log(e^{\lambda+\gamma}
+\sqrt{e^{2(\lambda+\gamma)}-1})$, $\gamma,\lambda\ge 0$, and obtain from (\[sle\]) $$v_\gamma:=v^{{\mathrm{que}}}_{\gamma}=v^{{\mathrm{ann}}}_{\gamma}=\sqrt{1-e^{-2\gamma}},\quad
\gamma\ge 0.$$ Thus, $\alpha_{\gamma}(1)=\beta_{\gamma}(1)\sim
\sqrt{2\gamma}$ and $v_{\gamma}\sim \sqrt{2\gamma}$ as $\gamma\downarrow 0$. The latter is (\[op1\]) in this special case.

The relation (\[op1\]) further supports an informal statement that when the potential $V$ is “small” (but not “sparse”!) both the quenched and annealed behavior in such a potential are well approximated by the behavior of the walk in a constant potential ${\mathbb{E}}(V)$.

*An open problem.* Theorems \[vque\], \[vann\], and asymptotics (\[op1\]) provide information about $v^{\mathrm{que}}_{p,M}$ and $v^{\mathrm{ann}}_{p,M}$ when $M\to\infty$ or $p\to 0$ or $M\to 0$. A more interesting and challenging question is to study surfaces formed by the speeds when $(p,M)\in[0,1]\times[0,\infty)$.

As the first step, fix $p\in(0,1)$ and consider $v^{\mathrm{que}}_{p,M}$ and $v^{\mathrm{ann}}_{p,M}$ as functions of $M$. It does not seem surprising that the annealed environment will typically be more sparse than the quenched one and thus the random walker will feel less need to quickly navigate towards the goal $y$. One expects that $v^{\mathrm{que}}_{p,M}$ is strictly increasing in $M$ for a fixed $p$ and that $v^{\mathrm{ann}}_{p,M}$ has a single maximum as $M$ changes from $0$ to $\infty$ (see Figure \[fig\]).

![Conjectured shape 