Figure 1

Comparison between the exact numerical diagonalization (circles), and different theoretical approaches: The stars (green) show the results of a renormalization group approach, where, if one considers the problem as finding the normal modes of a spring network, the springs and masses are renormalized. The solid line (red) depicts the analytical results of Eq. (2) for 2D and 3D, and Eq. (11) of 3 for 1D. The numerical results shown are for 1, 2 and 3 dimensions, with $N=1000$, averaged over 1000 realizations. The points were chosen in a line, square or box of side one, and $ϵ=0.1$, corresponding to $\xi ={10}^{-4}$ in 1D, $\xi =0.0032$ in 2D and $\xi =0.01$ in 3D. Notice that the graph shows the distribution density of the logarithm of the eigenvalue, which eliminates the governing $1/\lambda$ dependence, and allows us to observe clearly the deviations from it. To demonstrate this, the horizontal dashed line (blue) shows an exact $1/\lambda$ distribution. No fitting parameters are used.Reuse & Permissions

Figure 2

Demonstration of the structure of the eigenmodes. $N=5000$ points were chosen randomly and uniformly in a two dimensional box, with $ϵ=\xi /r_{nn}=0.1$. The eigenmodes become more delocalized as the eigenvalues approach zero, a condition made quantitative in Eq. (5), through the use of the RG approach. Eigenmodes are comprised of clusters of points, localized in real space.Reuse & Permissions