Eigenfunctions Localised on a Defect in High-Contrast Random Media

We study the properties of eigenvalues and corresponding eigenfunctions generated by a defect in the gaps of the spectrum of a high-contrast random operator. We consider a family of elliptic operators A^ε in divergence form whose coefficients possess double porosity type scaling and are perturbed on a fixed-size compact domain. The coefficients of A^ε are random variables generated, in an appropriate sense, by an ergodic dynamical system. Working in the gaps of the limiting spectrum of the unperturbed operator A^ˆε, we show that the point spectrum of A^ϵ converges in the sense of Hausdorff to the point spectrum of the homogenised operator A^hom as ε→0. Furthermore, we prove that the eigenfunctions of A^ε decay exponentially at infinity uniformly for sufficiently small ε. This, in turn, yields strong stochastic two-scale convergence of such eigenfunctions to eigenfunctions of A^hom.