Extreme statistics of complex random and quantum chaotic states

TL;DR

This paper derives exact formulas for the extreme intensity statistics of complex random states and applies these results to chaotic quantum systems, revealing how maximum and minimum intensities behave statistically.

Contribution

It provides the first exact analytical descriptions of extreme intensity statistics for complex random states and connects these results to chaotic quantum systems via random matrix theory.

Findings

Maximum intensity approaches Gumbel distribution slowly.

Minimum intensity rapidly approaches Weibull distribution.

Results are consistent with finite-N formulas for chaotic quantum maps.

Abstract

An exact analytical description of extreme intensity statistics in complex random states is derived. These states have the statistical properties of the Gaussian and Circular Unitary Ensemble eigenstates of random matrix theory. Although the components are correlated by the normalization constraint, it is still possible to derive compact formulae for all values of the dimensionality N. The maximum intensity result slowly approaches the Gumbel distribution even though the variables are bounded, whereas the minimum intensity result rapidly approaches the Weibull distribution. Since random matrix theory is conjectured to be applicable to chaotic quantum systems, we calculate the extreme eigenfunction statistics for the standard map with parameters at which its classical map is fully chaotic. The statistical behaviors are consistent with the finite-N formulae.