+ V(x;\omega)$ with Neumann boundary conditions. Suppose that the random variables $V(x;\omega)$ are IID, non-negative and non-constant, and that for some $\eta>0$ and $c>0$, all $u\in\DZ^d$ and all $\ell \ge \ell_0 >0$ $$\label{eq:cond.V.CT}
\pr{ |\ball_\ell|^{-1}\, \sum_{x\in \ball_\ell} V(x;\omega)\leq 2\eta }
\leq e^{-c|\ball_\ell|}.$$ Then for some $C>0$, $L_0\in\DN$ the lowest eigenvalue $E^{(\BN)}_0(\omega)$ of $H^{(\BN)}_{\ball_{L_0}(u)}(\omega)$ satisfies $$\pr{ E^{(\BN)}_0(\omega) \le  2C L_0^{-1/2} } \leq e^{-c|\ball_\ell(u)|^{1/4}}.$$ By Dirichlet–Neumann bracketing, the same bound holds true for the lowest eigenvalue of the LSO $H^{(\BD)}_{\ball_{L_0}(u)}(\omega)$ with Dirichlet boundary conditions.

A detailed discussion of the Lifshitz tails phenomenon, along with all ingredients of the proof of Lemma \[lem:CT.prob\], can be found, e.g., in Ref [@K07].

Using Lemma \[lem:CT\] and Lemma \[lem:CT.prob\], we come to the following

\[cor:CT.prob\] Under the assumptions of Lemma \[lem:CT.prob\], there exist $\tilde  L^{(3)}\in\DN$ and $C, c>0$ such that, for any $L_0\ge \tilde L^{(3)}$ and for some $\eta(L_0)>0$, $m(L_0) \ge CL_0^{-1/2}$, $$\label{eq:cor.CT.S}
\pr{ \exists\, E\in[E^0, E^0+ \eta(L_0)] :\, \ball_{L_0}(0) \text{ is } (E,m(L_0))-S }
\leq e^{-c L_0^{d/4}},$$ where $E^0$ is the lower edge of the spectrum of LSO $\Delta + V(x;\om)$ on the lattice $\DZ^d$.

The next statement is merely a reformulation of Lemma \[lem:NT.implies.nloc.IID\] for energies $E$ restricted to an interval $I\subset \DR$.

\[lem:CT.cor.loc\] 