Taking functions $f_t: \lambda\mapsto e^{it\lambda}$, $t\in\DR$, we see that the uniform bounds on dynamical localization in finite volumes $\ball_{L_k}(0)$, established in the previous sections, imply the dynamical localization on the entire lattice.

Theorem \[thm:DL.lattice.IID\] leads directly to the following, more traditional form of dynamical localization. Let ${\mathbf}{X}$ be a multiplication operator defined by $({\mathbf}{X}f)(x):= (1+\|x\|)f(x)$.

Under the assumptions of Theorem \[thm:DL.lattice.IID\], there exist $a,c>0$ such that for any finite subset $K\subset\DZ^d$ and any finite interval $I\subset\DR$ $$\label{eq:thm.DL.lattice.IID.propagator}
\operatorname{\mathbb{E}}\left[ \sup_{t\in{\mathbb}{R}} \;\left\| e^{a \ln^{1+c}{\mathbf}{X}} \, {\mathrm{e}}^{-{\mathrm{i}}tH(\omega)}
          P_{I }(H(\omega))
\operatorname{\mathbf{1}}_{K}\right\|\right] < \infty.$$

Exponential decay of eigenfunctions on the entire lattice {#ssec:SimpleSL.lattice}
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The dynamical localization is known to imply pure point spectrum, owing to RAGE theorem(s); see the original papers [@R69], [@AG73], [@E78] and their detailed discussion in [@CFKS87]. This allows to consider in Theorem \[thm:Main.SL\] below, from the beginning, a square summable (hence, bounded) eigenfunction $\psi$ on the lattice $\DZ^d$, avoiding a usual reference to a Shnol-type theorem stating that spectrally a.e. generalized eigenfunction is polynomially bounded. The general strategy goes back to [@DK89]; using the “Radial Descent lemma” (Lemma \[lem:SubH.IID\]) and making a small concession in probability bounds (which experts in MSA may notice) results in a shorter and more transparent 