$v^{\mathrm{que}}_{p,M}$ and $v^{\mathrm{ann}}_{p,M}$ in two regimes: “sparse” ($p\to 0$, $M$ is fixed) and “spiky” ($M\to\infty$, $p$ is fixed). Our main results are contained in the following two theorems. By $f(x)\sim g(x)$ as $x\to\alpha$ we mean that $\lim_{x\to \alpha}f(x)/g(x)=1$.

\[vque\] With $V_{p,M}$ as above, the quenched speed, $v^{que}_{p,M} $, satisfies $$\begin{aligned}
  \label{qp}&v_{p,M}^{{\mathrm{que}}}   \sim \frac{3p}{2} \text{ as } p\to
  0\ \text{uniformly in } M\in [M_0,\infty),\ \forall M_0>0;\\\label{qM}
&v_{p,M}^{{\mathrm{que}}} \sim \frac{3p}{2-p+2p^2} \text{ as } M\to\infty\
\text{locally uniformly in }p\in(0,1].\end{aligned}$$

\[vann\] With $V_{p,M}$ as above, the annealed speed, $v^{ann}_{p,M} $, satisfies $$\begin{aligned}
  \label{ap}&-\log v_{p,M}^{{\mathrm{ann}}}\sim \frac{2(e^M -1)}{p} \text{ as }
  p\to 0\ \text{uniformly in } M\in[M_0,\infty),\ \forall 
  M_0>0;\\\label{aM}&-\log v_{p,M}^{{\mathrm{ann}}}\sim \frac{2(1-p)e^M}{p}
  \text{ as } M\to\infty\ \text{locally uniformly in }p\in(0,1).\end{aligned}$$

[*Observe that in the “sparse” regime both the quenched and annealed speeds vanish but the latter does so at a dramatically higher rate. In the “spiky” regime the quenched speed converges to a positive constant while the annealed one vanishes extremely fast. This striking difference is a purely one-dimensional phenomenon. In dimensions two and higher the existence of the annealed asymptotic velocity is known ([@IV12b Theorem C]) but the existence of the quenched asymptotic velocity is still a largely open problem (see [@IV12a Section 1.2] and references 