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

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.