## 3 Results and Discussion
### Power Law Behavior of the Correlation Length Scale in Stationary State  
We have investigated the coherence length \(\xi\) with increasing pump power for various refractive indices and various sample extents. In Figure 5 we show the coherence length \(\xi\) for the scatterers arrangements of \(r_{scat}\ =\ 423.0\) nm, filling fraction of \(50\,\%\) and the samples extents of \(d\ =\ 40\,r_{scat}\ =\ 16.92\,\upmu\)m and \(d\ =\ 80\,r_{scat}\ =\ 33.84\,\upmu\)m. We show samples for \(n\ =\ 2.4\) (C), \(n\ =\ 2.3\) (ZnO) and \(n\ =\ 3.22\) (TiO\({}_{2}\)). The excitation wavelength is \(\lambda\ =\ 385.0\) nm; results are displayed for pump powers from \(0.2\,\gamma_{21}\) up to \(9.0\,\gamma_{21}\). We display the dependency of \(\xi\) with \(1/\sqrt{P}\) and we find a characteristic power law behavior for increasing pump power \(P\). Another, similar power law has been found in experiments of random lasers by [PERSON] et al. [2]. When we compare several materials we find the specific crossover behavior for the coherence length \(\xi\) for the samples of \(n\ =\ 2.4\) (C) and \(n\ =\ 3.22\) (TiO\({}_{2}\)) whereas the power law behavior is confirmed in either sample. The coherence length \(\xi\) for TiO\({}_{2}\) assumes values of up to \(\xi\ =\ 150.0\,\upmu\)m; thus the coherence volume has an extent in this case of up to \(\xi\ =\ 300.0\,\upmu\)m in stationary state.  
Figure 4: **(a)** Crossover of the coherence length \(\xi\) for various refractive indices. The crossover of the coherence length \(\xi\) for two materials \(n\ =\ 2.4\) (C) and \(n\ =\ 2.0041\) (ZnO) is displayed for the transport wavelengths \(\lambda\ =\ 580.0\) nm and \(\lambda\ =\ 620.0\) nm in-depth dependent of the random laser sample. The system parameters apart from the refractive index \(n\) and the transport wavelength \(\lambda\) are in either case identical, the scatterers radius is \(r_{scat}\ =\ 423.0\) nm. The samples finite extent in z-direction is of \(80\,r_{scat}\), the excitation strength \(P\ =\ 1.0\,\gamma_{21}\) is moderate. A principal crossover behavior of the length scale \(\xi\) is derived. We also find for the refractive index \(n\ =\ 2.4\) a crossover of the coherence lengths \(\xi\) near the sample’s boundaries for \(\lambda\ =\ 580.0\) nm and \(\lambda\ =\ 620.0\) nm in the stationary state. **(b)** Coherence length \(\xi\) compared to the scattering mean-free path \(l_{s}\) below threshold. The crossover of the coherence length \(\xi\) is shown in the samples center, \(Z=0\), parameters are identical to Figure 4a. Intuitively a crossover behavior of \(\xi\) is expected from the investigation of the scattering mean-free path \(l_{s}\), \(n\ =\ 2.4\) (C) (dashed line), \(n\ =\ 2.0041\) (ZnO) (solid line), for disordered arrangements of Mie scatterers below the laser threshold [31; 34]. Opposed to our results for \(l_{s}\) at the stationary state, see Figure 3a, we compare here to \(l_{s}\) in the weakly pumped case. In the spectral region between \(\lambda\ =\ 580.0\) nm and \(\lambda\ =\ 620.0\) nm we can confirm the crossover, which means that the Mie characteristics persist as a leading order effect under these specific conditions also while the system undergoes the lasing transition.