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We study numerically the spectrum of the non-Hermitian effective Hamiltonianthat describes the dipolar interaction of a gas of $N\gg 1$ atoms with theradiation field. We analyze the interplay between cooperative effects anddisorder for both scalar and vectorial radiation fields. We show that for densegases, the resonance width distribution follows, both in the scalar andvectorial cases, a power law $P(\Gamma) \sim \Gamma^{-4/3}$ that originatesfrom cooperative effects between more than two atoms. This power law isdifferent from the $ P(\Gamma) \sim \Gamma^{-1}$ behavior, which has beenconsidered as a signature of Anderson localization of light in random systems.We show that in dilute clouds, the center of the energy distribution isdescribed by Wigner's semicircle law in the scalar and vectorial cases. Fordense gases, this law is replaced in the vectorial case by the Laplacedistribution. Finally, we show that in the scalar case the degree of resonanceoverlap increases as a power law of the system size for dilute gases, butdecays exponentially with the system size for dense clouds.

Cooperative effects and disorder: A scaling analysis of the spectrum of the effective atomic Hamiltonian