\end{gathered}$$ This gives the desired statement with $c_2=q(1-e^{-M})^2/2$.

\[nc\] Let ${\varrho}=p/(1-p)$, $q=q({\varrho})$ be as defined in Corollary \[corbase\], and $c_2$ be the same as in Lemma \[lemfinal\]. Then in (\[23\]), $$Z_{0,y}^{-1}(1-p)^y\left(\frac{e^M}{{\varrho}}+1
  \right)\le c_2^{-1}(1-q)^y.$$

Final step: construction of $\Omega_y^1$ and $\Omega_y^2$ {#last}
---------------------------------------------------------

Throughout this subsection we suppose that, for a given environment $\omega$, the occupied sites in $(0,y)$ are $\{x_1,x_2,\dots,x_{n-1}\}$, where $0=x_0<x_1<\dots<x_{n-1}<y=:x_n$. As before, we set $r_i=x_i-x_{i-1}$, $i=1,2,\dots,n$.

Let ${\varrho}=p/(1-p)$. Fix an ${\varepsilon}>0$ and set $h=e^{2(1- {\varepsilon}/2) (e^M - 1)/{\varrho}}$. We claim that $h<EZ_1$ for all sufficiently small ${\varrho}$ ($M$ is fixed). Indeed, by (\[z\]) and Proposition \[propcalc\] $$EZ_1=\frac{\gamma(M){\varrho}(1-q({\varrho}))}{q({\varrho})},\
\text{where} \ \frac{1}{q({\varrho})}=e^{2(e^M-1)(1+o({\varrho}))/{\varrho}}\
\text{as }{\varrho}\to 0.$$

Let ${{\mathcal}A}_n:=\{\omega\in\Omega\,:\, N(\omega,y)=n,\
(R_1(\omega),R_2(\omega)\dots,R_n(\omega))\in A_n\}$, where $A_n$ was defined in (\[an\]). Then by Lemma \[corbond\] and Corollary \[nc\] we have that for all sufficiently small $p$ and large $y$ ($c_1$ and $c_2$ do not depend on $y$) $$\label{om}
  Q_{0,y}\left(\cup_{\{n:\ y/h\le n\le y\}}{{\mathcal}A}_{n}\right)\le
\frac{y}{c_2}\,e^{-c_1'(h)y}.$$

Now we are ready to construct $\Omega^1_y$. Let $\Omega_y^1=\cup_{\{n:\,1\le n<
  y/h\}}{{\mathcal}A}_n$. The bound (\[om\]) implies that $Q_{0,y}(\Omega_y^1)\ge
1/2$ for all sufficiently large $y$. Thus, (L1) is satisfied. Let $\omega\in\Omega_y^1$. By Lemma \[U\] and Cauchy-Schwarz inequality, $$E_{Q^\omega_{0,y}}\tau_y\ge \frac13 \sum_{i=1}^nr_i^2\ge
\frac{y^2}{3n}\ge \frac{y^2}{3(y/h)}\ge
\frac{hy}{3}\ge\frac{y}{3}\,e^{(1-{\varepsilon})K(p,M)}.$$ This gives us (L2) and the desired lower bound (\[lb\]).

The 