there exists $g_0 < \infty$ such that for all $g$ with $|g|\ge g_0$, any ball $\ball_{L_{0}}(u)$ and any $E\in\DR$ $$\begin{aligned}
\pr{ \ball_{L_{0}}(u) \text{ is } (E,m)\text{\rm-S} } &\le L_{0}^{-p}
\label{eq:L0.GF.IID} \\
\pr{ \ball_{L_{0}}(u) \text{ is } \mathnloc } &\le L_{0}^{-p}.
\label{eq:L0.loc.IID}\end{aligned}$$

Tunneling and localization in finite balls {#ssec:tun.and.loc.finite.balls}
------------------------------------------

\[lem:loc.and.NR.imply.NS.IID\] Let $E\in\DR$ and an ball $ \ball_L(u)$ be given. If $ \ball_L(u)$ is also $E$-NR, then it is $(E,m)$-NS.

The matrix elements of the resolvent $G_\ball(E)$ read as follows: \[eq:G.basis\] G\_(x,y;E) = \_[E\_j(H\_[\_L(u)]{})]{} . If $\operatorname{dist}(E, \sigma(H_\ball)) \ge e^{-L^\beta}$ and $\ln (2L+1)^d \le L^\beta$, then the property implies $$\bal
\displaystyle |G_\ball(x,y;E)|  &
& \le e^{-\gamma(m,L)\|x - y\| + L^\beta + \ln |\ball|}
 \le e^{-\gamma(m,L)L + 2L^\beta}.
 \hfill\qed
\eal$$

From this point on, we will work with a sequence of “scales” - positive integers $\{L_k, k\ge 0\}$, defined recursively by $L_{k+1} = [L_k^{\alpha}]+1$, $L_0>2$. In several arguments we will require the initial scale $L_0$ to be large enough.

\[lem:few.nloc.and.few.R.imply.nloc.IID\] There exists $\tilde L^{(1)}\in\DN$ such that for $L_k\ge \tilde L^{(1)}$

1.  if a ball $ \ball_{L_k}(u)$ is $E$-CNR and contains no pair of disjoint $(E,m)$-S balls of radius $L_{k-1}$, then it is also $(E,m)$-NS;

2.  if a ball $ \ball_{L_k}(u)$ is $\mathntun$ 