(Lemma \[lem54\]). Such comparison allows us to use standard large deviation bounds for product measures (Corollary \[corexp\]) and obtain sufficient control on the right-hand side of (\[23\]) to be able to construct $\Omega^1_y$ and $\Omega^2_y$ in the next subsection.

We use ${\varrho}$ below simply as a shorthand for $p/(1-p)$. Obviously ${\varrho}\sim p$ as $p\to 0$.

\[lem5basic\] There is a continuous function $\mu:(0,1)\to {\mathbb{R}}$ such that for every $q\in(0,1)$ $$\label{mu}
    \frac{q\log(1/q)}{2e^M(1-q)}\le \mu(q)\le  
\frac{q\log(1/q)}{2(e^M-1)(1-q)},$$ and $\mu^n(q)(1-e^{- M}) \le
  U_n(q) \leq \mu^n(q)$ for all $n\in{\mathbb{N}}$.

Let us fix an arbitrary $q\in(0,1)$ and drop it from the notation. It is obvious that $U_n\ge U_mU_{n-m}$ for $1\le m\le
  n$. This implies that the sequence $\log U_n$, $n\in{\mathbb{N}}$, is superadditive, and, thus, $$\label{un}
    \lim_{n\to\infty} \dfrac{\log U_n}{n}=\sup_n\frac{\log
    U_n}{n}=:\log \mu.$$ Therefore, $U_n\le \mu^n$ for all $n\in{\mathbb{N}}$.

For the lower bound, consider a killed random walk, which starts from the origin in an environment, such that all sites to the left from $0$ are empty. Let ${\widetilde}{U_n}$ be the probability that this walk reaches the $n$-th occupied site in $(0,\infty)$. Conditioning on the number of returns to the origin before reaching the $n$-th occupied site, we obtain $$\label{us}
    U_n\le {\widetilde}{U_n}\le
  \sum_{k=0}^\infty U_ne^{-Mk}=\frac{U_n}{1-e^{-M}}.$$ Notice that the sequence $\log{\widetilde}{U_n}$, $n\in{\mathbb{N}}$, is 