We investigate a family of N×N Euclidean random matrices S, whose
entries are sinc functions of the distance between points
independently sampled from a Gaussian distribution in three dimensions. This
random matrix model arises in the study of cooperative photon emission rates of
a random atomic cloud, initially excited by a laser in the linear regime. The
spectral properties of S, in the large-N limit, strongly depend on the
atomic cloud density. We show that in the low-density regime the eigenvalue
density of S has a nontrivial limit that only depends on the so-called
cooperativeness parameter b0, the only parameter of the model. For small
values b0≪1, we find that the limit eigenvalue density is approximatively
triangular. We also study the nearest-neighbour spacing distribution and the
eigenvector statistics. We find that, although S is a Euclidean random
matrix, the bulk of its spectrum is described by classical random matrix
theory. In particular, in the bulk there is level repulsion and the
eigenvectors are delocalized. Therefore, the bulk of the spectrum of S
exhibits the universal behaviour of chaotic quantum systems.Comment: 14 pages, 8 figure