  The study of the low temperature phase of spin glass models by means of Monte
Carlo simulations is a challenging task, because of the very slow dynamics and
the severe finite size effects they show. By exploiting at the best the
capabilities of standard modern CPUs (especially the SSE instructions), we have
been able to simulate the four-dimensional (4D) Edwards-Anderson model with
Gaussian couplings up to sizes $L=70$ and for times long enough to accurately
measure the asymptotic behavior. By quenching systems of different sizes to the
the critical temperature and to temperatures in the whole low temperature
phase, we have been able to identify the regime where finite size effects are
negligible: $\xi(t) \lesssim L/7$. Our estimates for the dynamical exponent ($z
\simeq 1/T$) and for the replicon exponent ($\alpha \simeq 1.0$ and
$T$-independent), that controls the decay of the spatial correlation in the
zero-overlap sector, are consistent with the RSB theory, but the latter differs
from the theoretically conjectured value.
