  The development of a Turing instability to a spatially modulated state in a
photoexcited electron-hole system is proposed as a novel signature of exciton
Bose statistics. We show that such an instability, which is driven by kinetics
of exciton formation, can result from stimulated processes that build up near
quantum degeneracy. In the spatially uniform 2d electron-hole system, the
instability leads to a triangular lattice pattern while, at an electron-hole
interface, a periodic 1d pattern develops. We analyze the mechanism of
wavelength selection, and show that the transition is abrupt (type I) for the
uniform 2d system, and continuous (type II) for the electron-hole interface.
