structure as $Y_2 = {
  \mathbf{p}
}^T \mathbb{I} ~{
  \mathbf{p}
}$ with $\mathbb{I}$ being the identity matrix. But, whereas the bilinear form $\mathbb{I}$ is a diagonal matrix, $R$ has vanishing diagonal entries and is dominated by non-diagonal ones. In analogy to the calculations that led to Eq. (\[equ:pTpAveraged\]), we find: $$\begin{aligned}
  \langle R_G^2 \rangle_{p_0} &= \frac{1}{N} \sum_{ij} \left( {
  \mathbf{Z}
}_i^T {
  \mathbf{Z}
}_j \right) \left( {
  \mathbf{X}
}_i^T R {
  \mathbf{X}
}_j\right) e^{-(\lambda_i + \lambda_j)t}.
  \label{equ:RG2Averaged}\end{aligned}$$ The equilibrium value in disorder average can be computed easily: $$\begin{aligned}
  \langle R_G^2 \rangle_\tau^{(eq)} &= \sum_{kl} d(k,l)^2 \langle p_k p_l \rangle_\tau^{(eq)}\\
    &= \langle p_k p_l \rangle_{\tau,k \neq l}^{(eq)} \sum_{kl} d(k,l)^2\end{aligned}$$ The two factors can now be evaluated independently: $$\begin{aligned}
  &\left\langle p_k p_l \right\rangle_{\tau, k\neq l}^{(eq)} = \frac{1}{2} \left\langle \left( p_k + p_l\right)^2 - p_k^2 - p_l^2 \right\rangle_{\tau, k\neq l}^{(eq)}\\
      &= \frac{1}{2}\left\langle \left( p_k + p_l\right)^2\right\rangle_{\tau, k\neq l}^{(eq)} - \left\langle p^2 \right\rangle_{\tau}^{(eq)}
      ~=~ \frac{\alpha}{N^2},\end{aligned}$$ where we have used the results of [@Sokolov2010]. The evaluation of $\sum_{k,l=1}^N d(k,l)^2$ yields $$\begin{aligned}
  \sum_{k,l=1}^N d(k,l)^2 \simeq \frac{N^4}{12},\end{aligned}$$ so that $$\begin{aligned}
  \langle R_G^2 \rangle_\tau^{(eq)} &\simeq \frac{\alpha N^2}{12}
  \label{equ:RG2eq}\end{aligned}$$ Fig. \[fig:RG2ofN\] shows the results of simulations of Eq. (\[equ:RG2Averaged\]) in the limit $t \rightarrow \infty$ for different $\alpha < 1$ 