We show that under $Q_{0,y}$ the distribution of gaps between occupied sites is “comparable” to a product of log-series distributions[^3] with the average gap $g_{\mathrm{ann}}$, where for fixed $M$ and small $p$ or fixed $p$ and large $M$ $$\label{gap} 
\log g_{\mathrm{ann}}\sim K(p,M):=2p^{-1}(1-p)(e^M-1).$$ For i.i.d. Bernoulli potentials the gap distribution is geometric, and we already have the result that the reciprocal of the quenched speed is proportional to the average gap between two occupied sites, which is now $g_{\mathrm{ann}}$. This observation together with (\[gap\]) leads to the limits (\[ap\]) and (\[aM\]).

We shall give a detailed proof of (\[ap\]). The proof of (\[aM\]) is very similar but easier and is omitted but we shall write all steps in such a way that they can be readily adapted to the case when $M\to\infty$ and $p$ is fixed. An informal derivation of the formula for $K(p,M)$ is given in the next subsection right after Corollary \[sp\].

Our goal will be to construct subsets of environments that are essential and on which the walk has the claimed speed behavior. More precisely, to obtain a lower bound on $-\log
v^{ann}_{p,M}$ we shall restrict $Q_{0,y}$ to environments $\Omega^1_y=\Omega^1_y({\varepsilon},p,M)$ with the following properties: for every ${\varepsilon}>0$ there 