  The SUE machine is used to extend by a factor of 1000 the time-scale of
previous studies of the aging, out-of-equilibrium dynamics of the
Edwards-Anderson model with binary couplings, on large lattices (L=60). The
correlation function, $C(t+t_w,t_w)$, $t_w$ being the time elapsed under a
quench from high-temperature, follows nicely a slightly-modified power law for
$t>t_w$. Very tiny (logarithmic), yet clearly detectable deviations from the
full-aging $t/t_w$ scaling can be observed. Furthermore, the $t<t_w$ data shows
clear indications of the presence of more than one time-sector in the aging
dynamics. Similar results are found in four-dimensions, but a rather different
behaviour is obtained in the infinite-dimensional $z=6$ Viana-Bray model. Most
surprisingly, our results in infinite dimensions seem incompatible with
dynamical ultrametricity. A detailed study of the link correlation function is
presented, suggesting that its aging-properties are the same as for the spin
correlation-function.
