so that \[eq:prob.S2.not.R2\] (2L\_[k+1]{} + 1)\^[2d]{} L\_[k+1]{}\^[- (1+b)\^[k-1]{}]{} &lt; L\_[k+1]{}\^[-p(1+b)\^[k+1]{}]{}, for $L_0$ (hence, $L_{k+1}$) large enough, provided that \[eq:cond.b\] - &gt; (1+b)\^[2]{}. Observe that for $b\in(0,1)$, $(1+b)^2 < 1+3b$ and $\frac{2d}{ p } > \frac{2d}{ p(1+b)^{k-1} }$. Therefore, if $p$ and $b$ fulfill the conditions , then is also fulfilled, yielding the last inequality in . With $\alpha = 4/3$, this reads as $p>16d$ and $b< \frac{1}{24} - \frac {2d}{3p}$. Therefore, with $L_0$ large enough, $$\qquad\qquad\qquad\qquad\qquad\qquad
\textstyle
\pr{ \CN_{k+1} }  <  \quart L_{k+1}^{-p(1+b)^{k+1}}.
\qquad\qquad\qquad\qquad\qquad\qquad{}_{{}_{\qed}}$$

\[thm:ind.loc.1p.IID\] For any $m>0$, $p>16d$ and $L_0>2$ there exist $g_0 < \infty$ and $b >0$ such that for any $g$ with $|g|\ge g_0$, all $k\ge 0$ and any ball $\ball_{L_{k}}(u)\subset \DZ^d$, \[eq:thm.ind.1p.loc.IID\] L\_[k]{}\^[-p(1+ b )\^k]{}.

The claim follows by induction from Lemma \[lem:induct.IID\] and Lemma \[lem:L.0.IID\].

In the next subsection we translate the results of previous sections into the language of eigenfunction correlators. Unlike the Fractional-Moment Method, the Multi-Scale Analysis does not provide exponential decay of EF correlators; usually one proves a polynomial decay with a fixed exponent. Owing to a stronger probabilistic bound of “unwanted” events in finite balls, of the form $L_k^{-p(1+b)^k}$, we will be able to prove a slightly stronger decay bound for 