  The present paper is devoted to new, improved bounds for the eigenfunctions
of random operators in the localized regime. We prove that, in the localized
regime with good probability, each eigenfunction is exponentially decaying
outside a ball of a certain radius, which we call the "localization onset
length". For $\ell>0$ large, we count the number of eigenfunctions having onset
length larger than $\ell$ and find it to be smaller than $\exp(-C\ell)$ times
the total number of eigenfunctions in the system. Thus, most eigenfunctions
localize on finite size balls independent of the system size.
