tunneling {#ssec:loc.tun}
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\[def:loc\] Given a sample $V(\cdot;\om)$, a ball $ \ball_L(u)$ is called $m$-localized (, in short) if any eigenfunction $\psi_j$ of the operator $H_{ \ball_L(u)}$ satisfies \[eq:def.mloc.IID\] |\_j(x)| |\_j(y)| e\^[-(m,L) x - y]{} provided that $\|x - y\|\ge L^{7/8}$. Otherwise, it is called $m$-non-localized ().

In terms of the parameter $\alpha$ (which is set to $4/3$ in Section \[sec:adapt.mixing\]), the lower bound on the distance $d(x,y) \ge L^{7/8}$ reads as $d(x,y) \ge L^{\frac{1+\rho}{\alpha} }$, where $\rho = (\alpha-1)/2 = 1/6$.

\[def:tun.IID\] Let real numbers $E$ and $m>0$ be given. A ball $\ball_{\ell^{\alpha}}(u)$ is called $m$-tunneling ($\mathtun$, in short), if it contains a pair of disjoint balls of radius $\ell$, and $m$-non-tunneling ($m$-NT), otherwise.

Observe that, unlike the property of $E$-resonance or $(E,m)$-singularity, the tunneling property **is not related to a specific value of energy $E$**, and even *a single* tunneling ball has a small probability. This allows to adapt a very simple idea due to Spencer [@Spe88] to a direct proof of localization in finite volumes.

Simplified scale induction {#sec:SimpleScaleInd}
==========================

Initial scale bounds
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The following lemma (more precisely, its first assertion ) goes back to the works [@FMSS85; @DK89].

\[lem:L.0.IID\] For any $L_0>2$, $m>0$ and $p>0$ 