  A fundamental assumption in our understanding of disks is that when the
Toomre Q>>1, the disk is stable against fragmentation into self-gravitating
objects (and so cannot form planets via direct collapse). But if disks are
turbulent, this neglects a spectrum of stochastic density fluctuations that can
produce rare, high-density mass concentrations. Here, we use a
recently-developed analytic framework to predict the statistics of these
fluctuations, i.e. the rate of fragmentation and mass spectrum of fragments
formed in a turbulent Keplerian disk. Turbulent disks are never completely
stable: we calculate the (always finite) probability of forming
self-gravitating structures via stochastic turbulent density fluctuations in
such disks. Modest sub-sonic turbulence above Mach number ~0.1 can produce a
few stochastic fragmentation or 'direct collapse' events over ~Myr timescales,
even if Q>>1 and cooling is slow (t_cool>>t_orbit). In trans-sonic turbulence
this extends to Q~100. We derive the true Q-criterion needed to suppress such
events, which scales exponentially with Mach number. We specify to turbulence
driven by MRI, convection, or spiral waves, and derive equivalent criteria in
terms of Q and the cooling time. Cooling times >~50*t_dyn may be required to
completely suppress fragmentation. These gravoturbulent events produce mass
spectra peaked near ~M_disk*(Q*M_disk/M_star)^2 (rocky-to-giant planet masses,
increasing with distance from the star). We apply this to protoplanetary disk
models and show that even minimum mass solar nebulae could experience
stochastic collapse events, provided a source of turbulence.
