Numerical study of the dynamics of some long range spin glass models

Abstract: We present results of a Monte Carlo study of the equilibrium dynamics of the one dimensional long-range Ising spin glass model. By tuning a parameter $\sigma$, this model interpolates between the mean field Sherrington-Kirkpatrick model and a proxy of the finite dimensional Edward-Anderson model. Activated scaling fits for the behavior of the relaxation time $\tau$ as a function of the number of spins $N$ (Namely $\ln(\tau)\propto N^{\psi}$) give values of $\psi$ that are not stable against inclusion of subleading corrections. Critical scaling ($\tau\propto N^{\rho}$) gives more stable fits, at least in the non mean field region. We also present results on the scaling of the time decay of the critical remanent magnetization of the Sherrington-Kirkpatrick model, a case where the simulation can be done with quite large systems and that shows the difficulties in obtaining precise values for dynamical exponents in spin glass models.