# Levy Statistics and the Glassy Behavior of Light in Random Fiber Lasers
## 3 Results
### Levy Statistics and Glassy Behavior in Er-RFL
Indeed, in the second Gaussian phase above the RL threshold, the system is in the regime with self-averaging of the gain [93]. In contrast, in the first Gaussian phase, it is still in the prelasing regime. Moreover, we also comment that the observed independence on the spatial dimensionality of the Levy character of intensity fluctuations finds support in the theoretical analysis based on Langevin equations, previously discussed in Section 3.2, which is considered to hold, irrespective of the spatial dimension [69; 82]. We further notice in Figure 6 that the abrupt decrease in the Levy index \(\alpha\) at the onset of RL behavior is closely related to the sharp linewidth reduction observed in Figure 2. Therefore, our results for the Er-RFL system corroborate the suggestion of [93], that the transition from the Gaussian to the Levy regime in RLs could be used as a universal identifier of the RL threshold (however, see also the comment below on the results of ref. [97]). In fact, we also included in Figure 6 the FWHM measurement, whose drastic change at the RL threshold nicely coincides with the first Gaussian-to-Levy statistical transition. For higher values of the input power, after reaching a minimum around the threshold, the index \(\alpha\) smoothly rises back to the Gaussian value \(\alpha=2\) achieved above the threshold. We finally discuss the connection of the above findings with the photonic spin-glass behavior recently reported in the Er-RFL system [68; 69]. As mentioned in Section 3.2, the characterization of the photonic RSB glassy phase in the RL regime requires the calculation of a specific two-point correlation function, which in the present context, can be defined either among the mode amplitudes, mode phases, or intensity fluctuations. In the latter case, that can be accessed experimentally, where the replica overlap parameter is defined as [42; 79]:  
\[q_{\gamma\beta}=\frac{\sum\limits_{k}\Delta_{\gamma}(k)\Delta_{\beta}(k)}{ \sqrt{\left[\sum\limits_{k}\Delta_{\gamma}^{2}(k)\right]\left[\sum\limits_{k} \Delta_{\beta}^{2}(k)\right]}}, \tag{8}\]  
where \(\gamma\), \(\beta=1,2,\ldots,N_{s}\) denote the replica labels (\(N_{s}=5000\) in ref. [68]), the average intensity at the wavelength indexed by \(k\) reads \(<I>(k)=\sum_{\gamma=1}^{N_{s}}I_{\gamma}(k)/N_{s}\), and the intensity fluctuation is  
Figure 6: Lévy index \(\alpha\) (circles) and FWHM (triangles) as a function of the normalized input pump power \(P/P_{th}\). Three statistical regimes of intensity fluctuations are observed: prelasing Gaussian with amplified spontaneous emission (\(\alpha=2\)), Lévy-type RL (\(0<\alpha<2\)) around the threshold, and Gaussian RL (\(\alpha=2\)) well above the threshold. The sharp decrease in \(\alpha\) at the first Gaussian-to-Levy transition nicely coincides with the abrupt change in FWHM at the RL threshold. (Reproduced with permission from ref. [68]). \(\Delta_{\gamma}(k)=l_{\gamma}(k)-<I>(k)\). In the photonic context, each output spectrum is considered a replica, i.e., a copy of the Er-RFL system under fairly identical experimental conditions. The PDF P(\(q\)) represents the distribution of values \(q=q_{\gamma\beta}\) of the mode-mode correlations between intensity fluctuations, Equation (8). In fact, as discussed in the previous subsection, P(\(q\)) is analogous to the Parisi order parameter in the RSB spin-glass theory of disordered magnetic systems [84]. In the present context, the distribution P(\(q\)) signalizes a photonic replica-symmetric paramagnetic prelasing regime if its maximum occurs exclusively at \(q_{\text{max}}=0\) (no RSB), or a RSB spin-glass RL phase if the maximum also assumes values \(\,1_{\text{$q_{\text{max}}$}\,1}\
eq 0\) (RSB) (see Section 3.2). Figure 7 shows a remarkable agreement between the onset of the Levy statistical regime of intensity fluctuations and the emergence of the RSB glassy RL phase in Er-RFL. Indeed, we observe that both the Levy and spin-glass behaviors are simultaneously present around the RL threshold.