Anderson localisation in infinite and finite resonator crystals with long-range interactions

Abstract

We establish Anderson localisation for random capacitance operators and quantify concentration of individual modes of finite capacitance matrices for clusters of subwavelength high-contrast resonators. The underlying geometry is fully periodic in dimensions $d=2,3$, with one resonator per cell, and the inverse material factors are independent and uniformly distributed. We retain the full capacitance interaction, whose coefficients decay exponentially with lattice separation. Under coordinate reflection symmetry, spectral and dynamical localisation hold near the upper spectral edge for every fixed positive disorder strength. At weak disorder $W$, we obtain an explicit localisation window of energy width proportional to $W$; prescribed positive intervals are localised at sufficiently large disorder. A variational boundary comparison and a material-variable eigenvalue-spacing estimate transfer localisation from truncated matrices to individual modes of the finite capacitance matrix. In a bounded sample of cell size $\varepsilon$, with probability tending to one, all except for $O(\varepsilon^{1-d}\log(1/\varepsilon))$ modes in the localisation interval concentrate within radii $O(\varepsilon\log(1/\varepsilon))$. The estimates are uniform for the contrast-dependent capacitance family at small contrast. We also prove a common limiting density of states for the truncated and finite matrices and illustrate concentration numerically.