lower edge of the random potential is given by $E^0 > -\infty$, localization can be established in a narrow interval $I = [E^0, E^0 + \eta]$, with sufficiently small $\eta>0$. Then representation can no longer be used; it is more convenient to start with the analysis of resolvents and modify the notion of “tunneling” balls as follows.

\[def:tun.low.E\] Let an interval $I\subset \DR$ and a number $m>0$ be given. A ball $ \ball_{\ell^{\alpha}}(u)$ is called $(m,I)$-tunneling ($(m,I)$-T) if, for some $E\in I$, it contains a pair of disjoint $(E,m)$-S balls of radius $\ell$. Otherwise, it is called $(m,I)$-non-tunneling ($(m,I)$-NT).

\[lem:CT\] Consider a lattice Schrödinger operator $H_\Lambda$ on a subset $\Lambda\subset \mathbf{Z}^d$. Suppose that for some $E\in\DR$, $\operatorname{dist}(E, \Sigma(H_\Lambda))\ge \eta>0$. Then for all $x,y\in\Lambda$ $$\label{eq:CT.BCH}
|(H - E)^{-1}(x,y)| \le 2\eta^{-1} e^{ - \frac{\eta}{5d} \|x-y\|}.$$

The proof of Combes–Thomas estimate [@CT] for lattice models can be found, e.g., in [@K07] where it is shown that one can take the exponent $\frac{\eta}{12d}$. A minor improvement of the argument from [@K07] allows to obtain $\frac{\eta}{5d}$. (In Eqn (11.10) from [@K07], one can use the inequality $\frac{2}{5}e^{1/5}< \half$ instead of a more generous bound $e^{1/12}< e^1 < 3$.)

\[lem:CT.prob\] Consider random LSO $H^{(\BN)}_{\ball_\ell(u)}(\omega) = \Delta 