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Photon statistics of a random laser

A general relationship is presented between the statistics of thermal radiation from a random medium and its scattering matrix S. Familiar results for black-body radiation are recovered in the limit S to 0. The mean photocount is proportional to the trace of 1-SS^dagger, in accordance with Kirchhoff's law relating emissivity and absorptivity. Higher moments of the photocount distribution are related to traces of powers of 1-SS^dagger, a generalization of Kirchhoff's law. The theory can be applied to a random amplifying medium (or "random laser") below the laser threshold, by evaluating the Bose-Einstein function at a negative temperature. Anomalously large fluctuations are predicted in the photocount upon approaching the laser threshold, as a consequence of overlapping cavity modes with a broad distribution of spectral widths.