  A general theory is presented for the spatial correlations in the intensity
of the radiation emitted by a random medium in thermal equilibrium. We find
that a non-zero correlation persists over distances large compared to the
transverse coherence length of the thermal radiation. This long-range
correlation vanishes in the limit of an ideal black body. We analyze two types
of systems (a disordered waveguide and an optical cavity with chaotic
scattering) where it should be observable.
