spatial *spread* of the state, the size of the spatial domain which the particle or excitation is confined to.

![Equilibrium value $\langle R_G^2\rangle_\tau^{(eq)}$ of the radius of gyration in dependence of the system size. In agreement with Eq. (\[equ:RG2eq\]), the simulation results (dots) follow $\langle R_G^2\rangle_\tau^{(eq)} \simeq \alpha N^2/12$  (dashed line).[]{data-label="fig:RG2ofN"}](RadiusofGyration2){width="\linewidth"}

It is defined as $$\begin{aligned}
  R_G^2 &= \sum_{kl} d(k,l)^2 p_k p_l,\label{equ:RG}\end{aligned}$$ with $d(k,l)$ denoting the distance between sites $k$ and $l$. Thus, the gyration radius gives the mean squared distance between the two diffusion particles. In a linear lattice with closed boundary conditions $d(k,l) = |k-l|$. In the case of a ring of size $N$ the distance is given by $$\begin{aligned}
  d(k,l) &= \begin{cases}
          |k-l|, &\text{if~} |k-l| \leq N/2\\
          N-|k-l|, &\text{else.}
            \end{cases}\end{aligned}$$

We say a state is localized if the radius of gyration is small. With the definition of the matrix $R \in \mathbb{R}^{N \times N}$ with the components $R_{kl} = d(k,l)^2$ we find: $$\begin{aligned}
   R_G^2 &= {
  \mathbf{p}
}^T R {
  \mathbf{p}
}.\end{aligned}$$

![Temporal evolution of $\langle Y_2\rangle_{p_0, \tau}$ () in comparison to $1/\langle R_G^2\rangle_{p_0, \tau}$ ([**+**]{}), calculated as average over 100 landscape realizations of size $N = 11^3$ and $\alpha = 0.37$. []{data-label="fig:RG2temporal"}](N1331Y2p0avGyration2){width="\linewidth"}

This has the same 