although in both cases the number of sites is $N = 11^3$.

In Fig. \[fig:spectrum\] we plotted the spectrum $\rho_\lambda$ for both Laplacians $L$ and $L^{(3D)}$ on linear and on double-logarithmic scales. Note that the cusp at $\lambda = 1$ is due to the fact that we use pure Pareto-distributed waiting times and it does not show up for analytic distributions as for example a one-sided Lévy-distribution with corresponding asymptotics. In any case, the behavior of $\rho_\lambda$ for large values of $\lambda$ does not seem interesting to us because we are looking for hints for the long-time behavior of the system which means we are concerned about small $\lambda$. The double-logarithmic plot of Fig. \[fig:spectrum\] reveals that for small values of $\lambda$, the spectrum $\rho_\lambda$ exhibits a power-law behavior. The exponent in the 3D case is in fact equal to $\alpha - 1$, as the dashed line in the double-logarithmic plot indicates. This is the exponent Bovier and Faggionato [@Bovier2005] determined for infinite-dimensional models. Note, that the dashed line in Fig. \[fig:spectrum\] is not a fit. The absolute value of the exponent for the 1D case is slightly larger than in 3D.

![Approximation of Eqs. (\[equ:Factorized\]) and (\[equ:1Dwith3Deigenvalues\]) for $N 