the vectors in Eq. (\[equ:Tk2\]) are actually built from the components of the eigenvectors of $L^T$. In the limit $N \rightarrow \infty$, however, it can be shown that the Lyapunov exponent $\gamma(\lambda)$ of both $L$ and $L^T$ is the same.)

As the Lyapunov exponent gives the rate of exponential growth in an (eigen-)vector, it is a standard measure for localization: $\gamma = 0$ means that the state is delocalized whereas a high Lyapunov exponent means strong localization. Considering this, it seems peculiar to us that the equilibrium state of the trap model should have a high value of participation ratio but a vanishing Lyapunov exponent $\gamma$:

![Participation ratio as a function of time for 1D and 3D systems. The parameters are: $N = 11^3$, $\alpha = 0.37$, the number of realizations is 100.[]{data-label="fig:Y21D3D"}](N1331a37real100Y2p0av1D3D){width="\linewidth"}

$$\begin{aligned}
  \gamma(0) &= \lim_{N \rightarrow \infty} \frac{1}{N} \left\langle \log \left\| \begin{pmatrix}
      0 & 1\\ -1 & 2
       \end{pmatrix}^N \right\|_1\right\rangle_\tau\\
       &= \lim_{N \rightarrow \infty} \frac{1}{N} \log \left\| \begin{pmatrix}
      1-N & N\\ -N & N+1
       \end{pmatrix}\right\|_1\\
       &= 0.\end{aligned}$$

Thus, the equilibrium state is essentially delocalized with respect to the its Lyapunov exponent. Since in all our simulations we consider finite systems, the measured Lyapunov exponent at $\lambda = 0$ 