different values of the density $\eta$ (ranging from $\eta \approx 2.5$ up to $\eta \approx 80$), the distributions of effective resonance frequencies are plotted with a different color, according to the bar on the right. The exact values chosen for the curves are emphasized by dotted white lines in the color-bar. The curves at $\eta \approx 2.5$ and $\eta \approx 3$ are calculated using the cylindrical system described in [Fig. \[fig:system\_representation\]]{}-a (with thickness $d =0.4\lambda_0$ and transverse radius $l_{\text{cyl.}}= 5\lambda_0$), while the others are evaluated for a sphere of radius $r=0.55\lambda_0$. []{data-label="fig:RG_plots"}](FINAL_RG_plots.pdf){width="\columnwidth"}

From the structure of $\mathcal M_{\text{pair}}$, we define the pairwise interaction parameter $\mathcal K_{ij}=\mathcal L_{ij}|G^{\text{near}}_{ij}|/(|\delta \omega_{ij}|+1)$. A large value of $\mathcal K_{ij}$ (which requires $\mathcal L_{ij}=1$) implies that the strong near-field interaction is able to strongly split the original resonances, accounting for possible differences in resonance frequencies of the pair. We thus identify the most strongly interacting pair as that with the largest value of $\mathcal K_{ij}$, as pictorially depicted in the first panel of [Fig. \[fig:RG\_plots\]]{}-b. Diagonalization of $\mathcal M_{\text{pair}}$ results in two, new interacting resonance frequencies $\omega_{\pm}={\langle \omega \rangle}_{ij}\pm \sqrt{\delta\omega_{ij}^2 + (G_{ij}^{\text{near}})^2}$. We can then obtain an approximately equivalent system by replacing the two original resonance 