# Levy Statistics and the Glassy Behavior of Light in Random Fiber Lasers
## 3 Results
### Theoretical Framework
Indeed, the experimental results obtained for the Er-RFL (see below) indicate [68] that the distribution of output intensities displays much larger variance and much stronger fluctuations close to the threshold, if compared with those below and above the threshold. In this sense, as argued below, the PDF of intensities is most properly described by the Levy \(\alpha\)-stable distribution [88] for long time measurements performed in CW pumped RLs, such as the Er-RFL system [68], or during an extensive number of shots in the case of pulsed RLs [89]. As we now turn the focus to the physical discussion on the PDF of intensity values, Equation (6), we initially observe that the analysis is strictly connected with the central limit theorem (CLT) and generalized CLT of statistics. These theorems determine the attraction of the PDF of the sum of a large number of random variables to one of the possible asymptotic stable distributions, namely the Gaussian or the Levy \(\alpha\)-stable family [88]. In the present photonic context, if the stochastic values assumed by the intensity \(I\) are identically distributed and uncorrelated over the long sequence of output spectra (or even if they present finite-time correlations), and if the second moment of the PDF \(P(I)\) is finite, then the CLT assures [88] that the intensity fluctuations are driven by the Brownian (Gaussian, normal) dynamics. On the other hand, if the second moment of \(P(I)\) diverges, the generalized CLT states [88] that the fluctuations are asymptotically governed by the Levy statistics. The continuous family of Levy \(\alpha\)-stable distribution is described [88] by the Fourier transform of the characteristic function defined in \(k\)-space,  
\[\overline{P}(k)=\exp\{-|ck|^{\alpha}[1-i\beta\text{sgn}(k)\Phi]+ik\
u\}. \tag{7}\]  
The Levy index \(\alpha\in(0,2]\) is the most important parameter, since it drives the magnitude of the intensity fluctuations. Indeed, whereas strong fluctuations with relevant deviations from the Gaussian behavior are associated with values in the range \(0<\alpha<2\), the Gaussian statistics with relatively weak fluctuations and the result of the CLT are recovered for the boundary value \(\alpha=2\). Therefore, Equation (7) can suitably describe both Gaussian and Levy statistical regimes, depending only on the value of the single parameter \(\alpha\). In other words, the parameter \(\alpha\), which can be experimentally determined from the direct analysis of the PDF \(P(I)\), effectively works as an indicator of the statistical regime (Gaussian or Levy) of intensity fluctuations. The other independent parameters describe the asymmetry or skewness of the distribution (\(\beta\in[-1,\,1]\)), location (\(\
u\in(-\infty,\infty)\)), and scale (\(c\in(0,\,\infty)\)), along with the function \(\Phi(k)=-(2/k)\ln{1}\,k\,|\,\) if \(\alpha=1\), whereas \(\Phi=\tan(\pi\alpha/2)\) if \(\alpha\
eq 1\). Though the Levy PDF, given by the Fourier transform of Equation (7), displays closed analytical form only for a few values of \(\alpha\) (e.g., the Cauchy distribution arises for \(\alpha=1\) and \(\beta=0\)), its large-\(I\) asymptotic behavior is power-law tailed, \(P(I)\sim I^{-\mu}\), with exponent \(\mu=1+\alpha\). Conversely, it is also true that random variables with power-law distribution are governed by the Levy PDF with \(\alpha=\mu-1\) if \(1<\mu<3\) (diverging second moment), and by the \(\alpha=2\) Gaussian statistics if \(\mu\geq 3\) (finite second moment) [88]. Therefore, if the PDF of intensities presents asymptotic power-law behavior, then the power-law exponent \(\mu\) also indicates the type of statistical regime (Gaussian or Levy) of the output intensity values. At this point, some words of caution are necessary in order to properly interpret the actual experimental data of the Er-RFL system under the statistical framework of the CLT and generalized CLT.