# Levy Statistics and the Glassy Behavior of Light in Random Fiber Lasers
## 3 Results
### Levy Statistics and Glassy Behavior in Er-RFL  
In this subsection, we focus on the statistical analysis of the experimental data of the Er-RFL system. We start by analyzing the intensity spectra. Figure 5a-c display 5000 spectra for each input power, from which the intensity fluctuations can be appreciated in the regimes below (Figure 5a), around (Figure 5b) and above (Figure 5c) the RL threshold [68]. According to the discussion in Section 3.2, the strong intensity fluctuations observed near the threshold suggest that the PDF of the output intensities can be described by the family of Levy \(\alpha\)-stable distributions, including the Levy statistical regime if \(0<\alpha<2\) and the Gaussian limit if \(\alpha=2\). Figure 5d-f portrays the distributions \(P(I)\) obtained from the data of Figure 5a-c, as well as the respective best fits to Equation (7) by applying the quantile-based method [93; 94]. Best-fit values of the parameters are summarized in Table 1. The values of \(\alpha\) are consistent with the Gaussian profiles (\(\alpha=2.0\)) shown in Figure 5d,f, respectively, below and above the threshold, and also with the Levy-like PDF (\(\alpha=1.3\)) around the threshold, observed in Figure 5e. The unit value of \(\beta\), indicating the maximum skewness of the distribution, in all cases reflects the asymmetry related to the positiveness of the intensity. Furthermore, the values of the location parameter \(\
u\) in the Gaussian regimes, \(\
u=0.858\) for \(P/P_{th}=0.6\) and \(\
u=0.682\) for \(P/P_{th}=1.8\), agree with the mean values, respectively, observed in Figure 5d,f. Indeed, the actual Gaussian distributions, which are equivalent to the \(\alpha=2\) Levy PDFs in Figure 5d,f, present the mean \(\
u\) and standard deviation \(\sqrt{2}c\), as theoretically predicted [88]. Interestingly, when comparing the Gaussian regimes below and above the threshold, we notice a considerable broadening of the PDF \(P(I)\) at \(P/P_{th}=1.8\), leading to a wider spread of intensities (Figure 5f). This result is associated with the larger second moment of the distribution and more intense fluctuations observed above the threshold. Those fluctuations still remain, however, much weaker than the ones measured in the crossover region. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Input Excitation Power** & \(\mathbf{\alpha}\) & \(\mathbf{\beta}\) & \(\mathbf{c}\) & \(\mathbf{\
u}\) \\ \hline \(P/P_{th}=0.6\) & 1.9 & 1.0 & 0.021 & 0.858 \\ \(P/P_{th}=1.2\) & 1.3 & 1.0 & 0.061 & \(-0.193\) \\ \(P/P_{th}=1.8\) & 2.0 & 1.0 & 0.091 & 0.682 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of best fit parameters to Equation (7) for the intensity distributions of Figure 5d–f [68]. Figure 5: **(a–c) 5000 intensity spectra of the Er-RFL system for each input power (a) below; (b) around; and (c) above the RL threshold; (d–f) Probability density functions (PDFs) \(P(I)\) of maximum intensities obtained from the data shown in (a–c). The best fits using Equation (7) are depicted in dotted lines and portrait Gaussian profiles (d) below and (f) above the threshold (\(\alpha=2.0\)); and (e) a Lévy distribution (\(\alpha=1.3\)) around the threshold. (Reproduced with permission from ref. [68]). **  
The main result confirming the Levy behavior of the output intensities of the Er-RFL system is shown in Figure 6. With a basis on ref. [68], we first notice that the variation of the Levy index \(\alpha\) as a function of the normalized input power clearly points to the presence of three distinct statistical regimes. Indeed, as \(P/P_{th}\) increases, the statistics of the output intensities progressively shift from the prelasing Gaussian (\(\alpha=2\)) to the Levy (\(0<\alpha<2\)) behavior around the threshold, and to the subsequent Gaussian (\(\alpha=2\)) regime deep in the RL phase. Noticeably, this sequence also resembles the statistical behavior of the output intensity of 3D bulk RLs [82; 89; 93; 95; 96]. We also notice that, as pointed in ref. [93], the second Gaussian regime is rather distinct from the first one.