We investigate, analytically near the dimension $d_{uc}=4$ and numerically in $d=3$, the non equilibrium relaxational dynamics of the randomly diluted Ising model at criticality. Using the Exact Renormalization Group Method to one loop, we compute the two times $t,t_w$ correlation function and Fluctuation Dissipation Ratio (FDR) for any Fourier mode of the order parameter, of finite wave vector $q$. In the large time separation limit, the FDR is found to reach a non trivial value $X^{infty}$ independently of (small) $q$ and coincide with the FDR associated to the the {it total} magnetization obtained previously. Explicit calculations in real space show that the FDR associated to the {it local} magnetization converges, in the asymptotic limit, to this same value $X^{infty}$. Through a Monte Carlo simulation, we compute the autocorrelation function in three dimensions, for different values of the dilution fraction $p$ at $T_c(p)$. Taking properly into account the corrections to scaling, we find, according to the Renormalization Group predictions, that the autocorrelation exponent $lambda_c$ is independent on $p$. The analysis is complemented by a study of the non equilibrium critical dynamics following a quench from a completely ordered state.
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