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The study of the low temperature phase of spin glass models by means of MonteCarlo simulations is a challenging task, because of the very slow dynamics andthe severe finite size effects they show. By exploiting at the best thecapabilities of standard modern CPUs (especially the SSE instructions), we havebeen able to simulate the four-dimensional (4D) Edwards-Anderson model withGaussian couplings up to sizes $L=70$ and for times long enough to accuratelymeasure the asymptotic behavior. By quenching systems of different sizes to thethe critical temperature and to temperatures in the whole low temperaturephase, we have been able to identify the regime where finite size effects arenegligible: $\xi(t) \lesssim L/7$. 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 thezero-overlap sector, are consistent with the RSB theory, but the latter differsfrom the theoretically conjectured value.

Spatial correlation functions and dynamical exponents in very large samples of 4D spin glasses