Wave-function extreme value statistics in Anderson localization

We consider a disordered one-dimensional tight-binding model with power-law decaying hopping amplitudes to disclose wave-function maximum distributions related to the Anderson localization phenomenon. Deeply in the regime of extended states, the wave-function intensities follow the Porter-Thomas distribution while their maxima assume the Gumbel distribution. At the critical point, distinct scaling laws govern the regimes of small and large wave-function intensities with a multifractal singularity spectrum. The distribution of maxima deviates from the usual Gumbel form and some characteristic finite-size scaling exponents are reported. Well within the localization regime, the wave-function intensity distribution is shown to develop a sequence of pre-power-law, power-law, exponential, and anomalous localized regimes. Their values are strongly correlated, which significantly affects the emerging extreme values distribution.