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We consider the non-equilibrium dynamics of disordered systems as defined bya master equation involving transition rates between configurations (detailedbalance is not assumed). To compute the important dynamical time scales infinite-size systems without simulating the actual time evolution which can beextremely slow, we propose to focus on first-passage times that satisfy'backward master equations'. Upon the iterative elimination of configurations,we obtain the exact renormalization rules that can be followed numerically. Totest this approach, we study the statistics of some first-passage times for twodisordered models : (i) for the random walk in a two-dimensional self-affinerandom potential of Hurst exponent $H$, we focus on the first exit time from asquare of size $L \times L$ if one starts at the square center. (ii) for thedynamics of the ferromagnetic Sherrington-Kirkpatrick model of $N$ spins, weconsider the first passage time $t_f$ to zero-magnetization when starting froma fully magnetized configuration. Besides the expected linear growth of theaveraged barrier $\bar{\ln t_{f}} \sim N$, we find that the rescaleddistribution of the barrier $(\ln t_{f})$ decays as $e^{- u^{\eta}}$ for large$u$ with a tail exponent of order $\eta \simeq 1.72$. This value can be simplyinterpreted in terms of rare events if the sample-to-sample fluctuationexponent for the barrier is $\psi_{width}=1/3$.

Statistics of first-passage times in disordered systems using backward master equations and their exact renormalization rules