is of order $\log(2N)/N$.

A double-logarithmic plot of the Lyapunov exponent for $N = 1331$ and $\alpha \in \{ 0.5, 0.75, 1.25, 2 \}$ is shown in Fig. \[fig:gammaOfECloseUp\]. For all four values of $\alpha$ the Lyapunov exponent $\gamma(0)$ is of order $\log(2N)/N$ (dashed line) which corresponds to zero in the infinite system size limit.

![image](SpectrumN1331a4.pdf){width=".48\linewidth"} ![image](SpectrumN1331a4loglog.pdf){width=".48\linewidth"}

The Lyapunov criterion of localization gives information which is strongly correlated with the one delivered by the gyration radius of single eigenstates. The inverse squared gyration radius for an eigenstate can be defined via $$\begin{aligned}
  \sigma_k^{-2} = \left( \frac{\sum_{i,j=1}^N d(i,j)^2 X_{ki}^2 X_{kj}^2}{\sqrt{\sum_{j=1}^N X_{kj}^4}}\right)^{-1}
  \label{equ:sigma-2},\end{aligned}$$ and $X_{kj}$ being the $j$th entry of the $k$th eigenvector. In Fig. \[fig:gammaOfsigma\] we plot $\gamma_k$ vs. $\sqrt{\sigma_k^{-2}}$ on double logarithmic scales. Strong correlation (linear on double logarithmic scales, i.e. a power-law dependence) between both is seen. A linear fit shows that $\log \gamma = 0.440 + 0.434 \cdot \log \sqrt{\sigma^{-2}}$. This means that the Lyapunov exponent and the gyration radius provide related information on the eigenstate localization: the eigenstates confined to a smaller region typically decay faster outside of it, and vice versa.

Spectral properties and dynamical localization {#subsec:spectrum}
----------------------------------------------

Now we turn to another important question, namely 