hint onto the main properties of the system responsible for dynamical localization.

![Temporal evolution of sample $\langle Y_2\rangle_{p_0}$ (dashed lines) and $\langle Y_2\rangle_{p_0, \tau}$ (dots) for $\alpha = 0.37$, $N = 11^3$ and the disorder average over 100 realizations. The light gray area gives the whole range of $\langle Y_2\rangle_{p_0}$ of all 100 realizations and the dark gray area depicts $\langle Y_2\rangle_{p_0,\tau} \pm \sigma$ with $\sigma$ being the standard deviation.[]{data-label="fig:Y2p0av"}](N1331a37real100Y2p0avWithSinglePaper){width="\linewidth"}

The Laplacian matrix
====================

Let us consider a 1D ring of $N$ sites and a realization $\{ \tau_k \}$ of waiting times according to the prescribed waiting time distribution $\rho_\tau$ of Eq. (\[equ:rhotau\]). Let ${
  \mathbf{p}
}(t) \in [0,1]^N$ be the vector whose components $\pi_k$ describe the probability to find the random walker at site $k$ at time $t$. Then the participation ratio $Y_2$ is simply ${
  \mathbf{p}
}^2 = {
  \mathbf{p}
}^T {
  \mathbf{p}
}$.

The temporal evolution of ${
  \mathbf{p}
}$ is given by the matrix $W$ of the transition rates $w_{k\rightarrow l}$ which are explicitly defined by Eq. (\[equ:GeneralRates\]). All eigenvalues of $W$ are nonpositive. In what follows we will refer to the Laplacian matrix (Laplacian) $L = -W$, with $(L)_{lk}=-w_{k\rightarrow l}$ and nonnegative eigenvalues.

The master 