Anderson localization in high-contrast media with random spherical inclusions

TL;DR

This paper demonstrates Anderson localization near band edges in high-contrast media with random spherical inclusions, revealing new insights into the spectral behavior of such composite materials.

Contribution

It is the first to prove Anderson localization near band edges in high-contrast random media, advancing understanding of the limiting spectral properties.

Findings

Anderson localization occurs near band edges.

Spectral behavior of high-contrast media is characterized.

Results rely on recent PDE unique continuation advancements.

Abstract

We study spectral properties of partial differential operators modelling composite materials with highly contrasting constituents, comprised of soft spherical inclusions with random radii dispersed in a stiff matrix. Such operators have recently attracted significant interest from the research community, including in the context of stochastic homogenization. In particular, it has been proved that the spectrum of these operators may feature a band-gap structure in the regime where heterogeneities take place on a sufficiently small scale. However, the nature of the limiting (as the small scale tends to zero) spectrum in the above setting is non-classical and not completely understood. In this paper we prove for the first time that Anderson localization occurs near band edges, thus shedding light on the limiting spectral behaviour. Our results rely on recent nontrivial advancements in quantitative unique continuation for PDEs, in combination with assumptions on the model that are standard in the Anderson localization literature, and which we plan to relax in future works.