Wavefunction extreme value statistics in Anderson localization

TL;DR

This paper investigates the statistical behavior of wavefunction maxima in a disordered one-dimensional model, revealing different regimes and distributions across extended, critical, and localized phases, with implications for Anderson localization.

Contribution

It introduces a detailed analysis of wavefunction maximum distributions across phases, including new scaling laws and the impact of correlations on extreme value statistics.

Findings

Wavefunction intensities follow Porter-Thomas distribution in extended states.

Maxima follow Gumbel distribution in extended states.

Distribution deviates from Gumbel at criticality with new scaling laws.

Abstract

We consider a disordered one-dimensional tight-binding model with power-law decaying hopping amplitudes to disclose wavefunction maximum distributions related to the Anderson localization phenomenon. Deeply in the regime of extended states, the wavefunction 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 wavefunction 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 wavefunction 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.