Lasing threshold of diffusive random lasers in three dimensions

We have numerically investigated the lasing threshold of three-dimensional diffusive random lasers composed of pointlike scatterers. The lasing threshold Gamma(T) is determined by the mode with the smallest decay rate. The dependence of Gamma(T) on the number of scatterers N (system size) is shown to follow the power law Gamma(T)proportional to N-2/3. This result, which reflects the diffusion law for the lasing threshold, is shown to apply even when the system is very close to the Anderson localization transition. We suggest that this result corroborates the scenario in which extended modes are at the origin of random lasing in the diffusive regime, as revealed by recent experiments.