[en] As was recently shown in Abumwis et al (2020 Phys. Rev. Lett. 124 193401), many eigenstates of a random Rydberg gas with resonant dipole–dipole interactions are highly delocalized. Although the high degree of delocalization is generic to various types of power-law interactions and to both two and three-dimensional systems, in their detailed aspects the coherence distributions are sensitive to these parameters and vary dramatically between different systems. We calculate the eigenstates of both two and three-dimensional gases and quantify their delocalization throughout the atoms in the gas using a coherence measure. By contrasting the angular dependence of the dipole–dipole interaction with an isotropic interaction we obtain additional information about the generic physical principles underlying random interacting systems. We also investigate the density of states and microwave absorption spectra to obtain information about the types of measurements where these delocalized states play a role, and to check that these delocalized eigenstates are robust against various types of perturbation. (paper)