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Anderson localization in high-contrast media with random spherical inclusions

Matteo Capoferri, Matthias Täufer

TL;DR

This work analyzes Anderson localization in high-contrast random media consisting of randomly sized soft spherical inclusions in a stiff matrix, within the double-porosity resonant regime. It develops a rigorous framework based on scale-free quantitative unique continuation to establish Wegner and initial-scale estimates, then applies a bootstrap multiscale analysis to prove localization near band edges and the existence of spectral gaps for small ε. The results demonstrate pure-point spectrum with exponentially decaying eigenfunctions and dynamical localization, linking micro-resonant structure to macroscopic spectral behavior in stochastic high-contrast homogenization. Overall, the paper advances understanding of localization phenomena in random high-contrast PDE models and provides a pathway for further relaxation of regularity and randomness assumptions.

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.

Anderson localization in high-contrast media with random spherical inclusions

TL;DR

This work analyzes Anderson localization in high-contrast random media consisting of randomly sized soft spherical inclusions in a stiff matrix, within the double-porosity resonant regime. It develops a rigorous framework based on scale-free quantitative unique continuation to establish Wegner and initial-scale estimates, then applies a bootstrap multiscale analysis to prove localization near band edges and the existence of spectral gaps for small ε. The results demonstrate pure-point spectrum with exponentially decaying eigenfunctions and dynamical localization, linking micro-resonant structure to macroscopic spectral behavior in stochastic high-contrast homogenization. Overall, the paper advances understanding of localization phenomena in random high-contrast PDE models and provides a pathway for further relaxation of regularity and randomness assumptions.

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.

Paper Structure

This paper contains 7 sections, 16 theorems, 83 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Statement of the problem
  3. Main results
  4. Gaps in the spectrum of $\mathcal{A}_\omega^\varepsilon$
  5. Wegner estimate
  6. Proof of initial scale estimate
  7. Multiscale analysis

Key Result

Theorem 3.1

Suppose that Assumptions assumption:bounded_density and assumption:epsilon_and_omega hold, and let $E_0 > 0$ be a lower band edge of $\sigma(\mathcal{A}_{\omega}^\varepsilon)$. Then there exist $\kappa>0$ and $E_+ > E_0$ such that, if Assumption assumption:thinness_density with exponent $\kappa$ is

Figures (3)

  • Figure 1: The sets $\mathcal{I}_\omega^\varepsilon$ (white), $\mathcal{L}_\omega^\varepsilon$ (light grey) and $\mathcal{M}_\omega^\varepsilon$ (dark grey) in two dimensions (left). On the right-hand side, the same sets for a Delone-configuration.
  • Figure 2: Geometric visualisation of Theorem \ref{['theorem gaps are protected']}.
  • Figure 3: Illustration of the sets $\Lambda_L(x)$ and $\Gamma_L(x)$ in Definition \ref{['def:Gamma']}.

Theorems & Definitions (40)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.5
  • Remark 2.6
  • Definition 2.8: Lower and upper band edges
  • Theorem 3.1: Anderson localization
  • Theorem 3.2: Wegner estimate
  • Theorem 3.3: Initial scale estimate (ISE)
  • Remark 3.4
  • Theorem 3.5: Strong sub-exponential Hilbert-Schmidt-kernel decay
  • ...and 30 more