$L^*>0$ and $\beta' \in(0, \beta)$ such that

1.  for any $E\in I$, any $L\geq L^*$ and any ball $\ball_{L}(x)$ $$\label{eq:W1.CNR.IID}
    \pr{ \text{ $\ball_{L}(x)$ is not $E$-\rm{CNR} } }
    \le e^{-L^{\beta'}};$$

2.  for all $L\geq L^*$ and any pair of disjoint balls $\ball_{L}(x)$, $\ball_{L}(y)$ $$\label{eq:W2.IID}
    \pr{ \exists\, E\in I\, \text{ $\ball_{L}(x)$ and $\ball_{L}(y)$  are $E$-\rm{PR} } }
    \le e^{-L^{\beta'}}.$$

In the case where the random potential is bounded one can take the interval $I$ containing the entire spectrum of the operator $H(\om)$.

\[def:SubH.IID\] Let be given two integers $L\ge \ell\ge 0$ and a number $0 < q <1$. Consider a finite connected subgraph $\ball\subset\DZ^d$. A function $f:\, \ball\to\DR_+$ will be called $(\ell,q)$-subharmonic in a ball $\ball_L(u)\subsetneq\ball$ if for any $\ball_\ell(x)\subset\ball_L(u)$ one has \[eq:def.l.q.subh.IID\] f(u) q f(y).

The relevance of this notion for the analysis of localization properties of eigenfunctions is explained by Lemma \[lem:cond.SubH.IID\] below. Analytic (deterministic) bounds given by Lemma \[lem:SubH.IID\] and Lemma \[lem:BiSubH.IID\] can be considered as a convenient alternative to the well-known technique going back to [@FMSS85], [@DK89], also based on the GRI.

\[lem:SubH.IID\] Let $\ball$ be a finite connected subgraph of the lattice $\DZ^d$ and $f: \ball\to\DR_+$ is an $(\ell,q)$-subharmonic function which is $(\ell,q)$-suharmonic in $\ball_L(u)\subsetneq\ball$, with 