Universality of the Wigner time delay distribution for one-dimensional random potentials

TL;DR

This paper demonstrates that the distribution of Wigner time delay in one-dimensional random potentials is universal at high energies or weak disorder, with analytical and numerical evidence supporting this universality and exploring deviations at low energies.

Contribution

It establishes the universality of the Wigner time delay distribution in certain regimes and provides analytical, numerical, and physical insights into its behavior and deviations.

Findings

Distribution is universal at high energy or weak disorder.

Analytical results match extensive numerical simulations.

Deviations from universality occur at low energies.

Abstract

We show that the distribution of the time delay for one-dimensional random potentials is universal in the high energy or weak disorder limit. Our analytical results are in excellent agreement with extensive numerical simulations carried out on samples whose sizes are large compared to the localisation length (localised regime). The case of small samples is also discussed (ballistic regime). We provide a physical argument which explains in a quantitative way the origin of the exponential divergence of the moments. The occurence of a log-normal tail for finite size systems is analysed. Finally, we present exact results in the low energy limit which clearly show a departure from the universal behaviour.