Random M\"{o}bius Maps: Distribution of Reflection in Non-Hermitian 1D Disordered Systems

TL;DR

This paper explores the statistical behavior of reflection in one-dimensional disordered systems using random Möbius transformations, revealing conditions for perfect absorption and characterizing the distribution's tails and wave penetration.

Contribution

It introduces a novel analytical approach using Möbius transformations to study reflection properties and absorption conditions in non-Hermitian disordered systems.

Findings

Distribution support and absorption conditions explicitly determined

Lifshits-like tails at distribution boundaries evaluated

Wave penetration extent quantified via Lyapunov exponent

Abstract

Using the properties of random M\"{o}bius transformations, we investigate the statistical properties of the reflection coefficient in a random chain of lossy scatterers. We explicitly determine the support of the distribution and the condition for coherent perfect absorption to be possible. We show that at its boundaries the distribution has Lifshits-like tails, which we evaluate. We also obtain the extent of penetration of incoming waves into the medium via the Lyapunov exponent. Our results agree well when compared to numerical simulations in a specific random system.