  Powerful incoherent laser pulses can propagate in focusing Kerr media much
longer distances than can coherent pulses, due to the fast phase mixing that
prevents transverse filamentation. This distance is limited by 4-wave
scattering, which accumulates waves at small transverse wavenumbers, where
phase mixing is too slow to retain the incoherence and thus prevent the
filamentation. However, we identify how this theoretical limit can be overcome
by countering this accumulation through transverse heating of the pulse by
random fluctuations of the refractive index. In these new regimes, the laser
pulse propagation distances are significantly extended, making feasible a new
class of random lasers, in particular, ultra-powerful random lasers in plasmas.
