the third panel of [Fig. \[fig:RG\_plots\]]{}-b, is that the original, homogeneous system can be mapped to an optically equivalent system that is inhomogeneously broadened, with a smooth probability distribution of resonance frequencies $P(\omega_{\text{eff}})$.

Before describing more the characteristics of $P(\omega_{\text{eff}})$, we discuss a few subtle but important points associated with the RG process. First, we point out the historic work of [@Levitov1990DelocalizationInteraction], which used RG to understand the properties of permanent, *static* dipoles, which only experience a near-field $1/\rho^3$ interaction. Given only a near-field interaction in three dimensions, the interaction of a dipole with its nearest neighbor is then indeed dominant. However, we have a qualitatively different system, of driven, radiating dipoles. Naively then, a similar argument considering the $1/\rho$ far field would suggest that atoms within a shell of radius $\rho$ and $\rho+d\rho$ of one atom at the origin would contribute an interaction strength of $\sim \rho d\rho$, such that the *furthest* atoms actually play the strong role. We argue that an RG process based on the near field is still the correct prescription, as the index should be a local property. Instead, the apparent “dominance” of the far field simply reflects the fact that the macroscopic geometry of 