and system sizes $N$ and the comparison to Eq. (\[equ:RG2eq\]).

The important information is that the equilibrium value of the radius of gyration grows with the system size whereas the equilibrium value of $\langle Y_2 \rangle_\tau$ becomes independent of $N$ (for $N$ large enough), i.e. the states localized according to the $Y_2$ criterion are delocalized with respect to their gyration radius.

The reason is quite clear. If a site attracts a considerable amount of probability in the equilibrium state ${
  \mathbf{p}
}^{(eq)} = (\tau_1, \tau_2, \ldots , \tau_N)^T/\sum_k\tau_k$, let us call it a “deep trap”. If there is only one deep trap in the landscape, $Y_2$ is large, and $R_G$ is small. Many landscape realizations do contain *more than one* deep trap, in which case $Y_2$ is still large, but $R_G$ might be large as well, being of the size of the distance between the deep traps. The expectation of the distance between the deep traps grows with the system size and thus the radius of gyration does also grow, whereas the participation ratio is independent of the deep traps’ positions.

Apart from the equilibrium value, we can also compare the temporal evolution of $1/\langle R_G^2\rangle_{p_0, \tau}$ to the one 