\sum_{ij} G_{ij} e^{-(\lambda_i + \lambda_j)t}.
   \label{equ:pTpAveraged}\end{aligned}$$ where the elements $$\begin{aligned}
  G_{ij} &= ({
  \mathbf{Z}
}_i^T {
  \mathbf{Z}
}_j)({
  \mathbf{X}
}_i^T {
  \mathbf{X}
}_j)\label{equ:DefG}{
  \parbox[0pt][2em][c]{0cm}{}
}\\
    &= ({
  \mathbf{Q}
}_i^T S^2 {
  \mathbf{Q}
}_j)({
  \mathbf{Q}
}_i^T S^{-2} {
  \mathbf{Q}
}_j){
  \parbox[0pt][2em][c]{0cm}{}
}\nonumber,  \end{aligned}$$ define a new matrix $G$ which is the element-wise product of the Gramian matrices corresponding to the vector sets $\{ {
  \mathbf{Z}
}_1, {
  \mathbf{Z}
}_2, \ldots, {
  \mathbf{Z}
}_N \}$ and $\{ {
  \mathbf{X}
}_1, {
  \mathbf{X}
}_2, \ldots, {
  \mathbf{X}
}_N \}$. Thus, the temporal development of $\langle Y_2\rangle_{p_0}$ depends on $G_{ij}$ in relation to the characteristic decay times $1/(\lambda_i + \lambda_j)$. As we proceed to show, these are the properties of $G$ which dominate the dynamical localization.

Taking the temporal limit of Eq. (\[equ:pTpAveraged\]) in a finite system and averaging over disorder, we can also reproduce the finite-size equilibrium value $\langle Y_2 \rangle^{(eq)}_\tau$ as it was given in [@Bertin2002; @Derrida1997]: $$\begin{aligned}
  \left\langle \lim_{t\rightarrow \infty} \langle Y_2\rangle_{p_0} \right\rangle_\tau &= \frac{1}{N} \left\langle\left( {
  \mathbf{Z}
}_1^T {
  \mathbf{Z}
}_1 \right)
                      \left( {
  \mathbf{X}
}_1^T {
  \mathbf{X}
}_1\right)\right\rangle_\tau\nonumber\\
      &= \left\langle\frac{1}{\sum_k \tau_k} \left( {
  \mathbf{X}
}_1^T {
  \mathbf{X}
}_1\right)\right\rangle_\tau\nonumber\\
      &= \left\langle\frac{\sum_l \tau_l^2}{\left(\sum_k \tau_k\right)^2}\right\rangle_\tau\simeq 1-\alpha,\end{aligned}$$ see [@Sokolov2010].

Eq. (\[equ:pTpAveraged\]) comes in 