so-called Geometric Resolvent Inequality) where $
C_\ell = |\pt  \ball_\ell(u)|  \le C(d) \ell^{d-1}.
$ The latter bound, less precise than the former, is sufficient for the MSA. Similarly, for the eigenfunctions $\psi$ of operator $H_{\ball_L(w)}$ with eigenvalue $E \not\in \Sigma(H_{\ball_\ell(u)})$ we have \[eq:GRI.EF\] | (x) | ( C\_ \_[v: d(x,v)=]{} |G\_[\_(x)]{}(x,v;E) | ) \_[y: d(u,y)+1]{} | (y) |, x\_(u). The same inequalities remain valid in the situation where $\ball_\ell(u)$ is not entirely a subset of $\ball_L(w)$. In such a case, however, one has to replace the ball $\ball_\ell(u)$ and its boundaries by their intersections with the set $\ball_L(w)$; see the details in Appendix.

Decay of Green functions and eigenfunctions  {#sec:MSA.IID}
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“Radial descent” bounds for Green functions and eigenfunctions {#ssec:radial.descent}
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Now we fix the parameter $\beta\in(0,1)$; for our purposes, it is convenient to take $\beta > 1/4$, so that one can set the exponent (figuring in Lemma \[lem:WS1.IID\]) $\beta'=1/4$.

\[def:CNR.IID\] Given a sample $V(\cdot;\om)$, a ball $\ball_L(u)$ is called

1.  $E$-non-resonant ($E$-NR) if $\|G_\ball(E;\om)\|\leq e^{+L^{\beta} }$, and $E$-resonant ($E$-R), otherwise;

2.  completely $E$-non-resonant ($E$-CNR) if it does not contain any $E$-R ball $\ball_\ell(u)$ with $\ell\ge L^{1/\alpha}$, and $E$-partially resonant ($E$-PR), otherwise.

\[lem:WS1.IID\] Fix a finite interval $I\subset\DR$. There exist 