## 3 Results and Discussion
### Material-Dependency of the Mie Characteristics in Multiple Scattering Random Lasers  
We know from the results for the scattering mean-free path \(l_{s}\) for photonic transport in the weak excitation regime that a crossover of \(l_{s}\) is expected for various Mie resonators of a differing refractive index \(n\) and thus a differing permittivity \(e_{scat}\). The result for the calculation of the scattering mean-free path [34] shows smooth but pronounced dips when the Mie resonance condition for the single scatterer is approached and finally met. This Mie resonance condition depends on the refractive index of the scatterer; its position will thus shift in the energy spectrum for various materials and identical scatterer parameters otherwise. Under an ideal choice of parameters an almost perfect point symmetry with respect to the crossover of the scattering mean free path \(l_{s}\) for two different refractive indices is possible. Here we refer to the literature, references [1; 2], where the gain mechanism in solid state random lasing is vividly discussed with a focus on two possible mechanisms, microscopic gain and gain due to artificial but randomly formed laser cavities in the sense of a build up of correlated chains of single scatterers. Whereas for the case of microscopic gain the properties of the single Mie scatterer should change with respect to the resulting change of the refractive index, in the case of artificial but randomly formed laser cavities the properties of the single scatterer should remain rather unchanged. Figure 3: (**a**) Scattering mean-free path \(l_{s}\) in the laser active system at threshold. In-depth dependent scattering mean-free path \(l_{s}\,=\,\frac{1}{2\text{Im}[\sqrt{q^{2}+l}\text{Im}\Sigma(\omega)]}\), for details see [34], in dependency to the excitation strength \(P\). Parameters are identical to those of Figure 2. We find an in-depth dependent renormalization of \(l_{s}\) which is increased with strength \(P\). Both quantities, the mean-free path \(l_{s}\) and the diffusion coefficient \(D(\Omega\,=\,0;\,Z)\), Figure 3b, are self-consistently derived and thus show quantitatively a different behavior at the sample boundaries \(Z\,=\,\pm 0.5d\), rather than at \(Z\,=\,0\). At the open boundaries both characteristics \(l_{s}\) and \(D(\Omega\,=\,0;\,Z)\) are increased. This behavior comes, however, along with the huge but inverse effect in the characteristics of the coherence volume, compare Figure 2. (**b**) Diffusion coefficient \(D\) of the random laser in stationary state. In-depth dependent diffusion coefficient \(D(\Omega\,=\,0;\,Z)\) as a material characteristic of the random laser arrangement at the laser threshold in the stationary state. Parameters are identical to Figures 2 and 3a. In the excitation regime \(P\,<\,1.0\,\gamma_{21}\) a moderate but existing dependency with respect to the external excitation and the position in-depth is found. In the regime for excitation above the microscopic laser threshold \(P\,<\,9.0\,\gamma_{21}\) we find an increasing deviation of the value of the diffusion coefficient D at the sample boundary of more than 7% as compared to the transport wavelength. This deviation is only correlated to the absolute excitation power and to the open boundaries of the sample where 85 % of all reemitted photons shall be lost, whereas the in-depth sample of all photons multiplies scattering. Both results of \(l_{s}\) and of \(D\), which is finite, are signs of the absence of Anderson localization [46]. This discussion for sure will be finally decided by novel experiments, which check the spatial extent of random laser modes in stationary state systematically by varying the geometric properties of the laser sample.