Analytical solution for the Lévy-like steady-state distribution of intensities in random lasers

We derive analytically the Lévy-like steady-state distribution with exponential tempering of emission intensities in random lasers. Our approach is based on the Langevin and associated Fokker-Planck equations describing the dynamics of the amplitudes of the resonance modes in a cavity with a disordered nonlinear dielectric medium. The reported results fully agree with the experimental characterization of the prelasing, Lévy-like, and self-averaged Gaussian lasing regimes in a random laser system as a function of the pump energy and disorder strength, as well as with the recent suggestion of the Lévy exponent $\alpha$ as a universal identifier of the random lasing threshold.