Rogue-wave statistics in Anderson chains

Abstract

The 1D Anderson model featuring uncorrelated diagonal disorder is considered. The wavefunction statistics associated to transitions between distinct locations is analyzed. In the presence of mild disorder, the local squared wavefunctions, that is occupation probabilities, obey exponential statistics. When disorder is high, amplitudes measured near the input site are well described by Rician distributions, a form of sub-exponential statistics, due to the influence of strongly localized modes. This results in a reduced likelihood of rogue wave events. When the statistics is taken over various disorder realizations or locations, the lack of knowledge over the rate of the exponential processes acting locally yields long-tailed distributions. As a consequence, rogue waves become more frequent at locations closer to the input for increasing disorder strength. Our findings can be used to assess the occurrence of extreme events as well as the degree of localization over a broad class of disordered models.