  We study the crossover between the diffusive and quasi-ballistic regimes of
random lasers. In particular, we compare incoherent models based on the
diffusion equation and the radiative transfer equation (RTE), which neglect all
wave effects, with a coherent wave model for the random laser threshold. We
show that both the incoherent and the coherent models predict qualitatively
similar thresholds, with a smooth transition from a diffuse to a
quasi-ballistic regime. The shape of the intensity distribution in the sample
as predicted by the RTE model at threshold is also in good agreement with the
coherent model. The approximate incoherent models thus provide useful
analytical predictions for the threshold of random lasers as well as the shape
of the random laser modes at threshold.
