Anderson localization in the subwavelength regime

Random media are ubiquitous in both natural and artificial structures and there are many important problems related to understanding their wave-scattering properties. In particular, the phenomenon of wave localization in random media, known as Anderson localization in certain settings, has proved difficult to understand, particularly in physically derived models and systems with long-range interactions. In this article, we show that the scattering of time-harmonic waves by high-contrast resonators with randomly chosen material parameters reproduces the characteristic features of Anderson localization and its properties can be understood using our asymptotic results. In particular, we show that the hybridization of subwavelength resonant modes is responsible for both the repulsion of energy levels as well as the widely observed phase transition, at which point eigenmode symmetries swap and very strong localization is possible. We derive results from first principles, using asymptotic expansions in terms of the material contrast parameter, and obtain a characterisation of the localized modes in terms of Laurent operators and generalized capacitance matrices. This model captures the long-range interactions of the wave-scattering system and provides a rigorous framework to explain the exotic phenomena that are observed.