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We develop an ab initio analytic theory of random lasing in an ensemble ofatoms that both scatter and amplify light. The theory applies all the way fromlow to high density of atoms. The properties of the random laser are controlledby an Euclidean matrix with elements equal to the Green's function of theHelmholtz equation between pairs of atoms in the system. Lasing threshold andthe intensity of laser emission are calculated in the semiclassicalapproximation. The results are compared to the outcome of the diffusion theoryof random lasing.

Euclidean matrix theory of random lasing in a cloud of cold atoms