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 $mathcal{A}^varepsilon$ in divergence form whose coefficients possess double porosity type scaling and are perturbed on a fixed-size compact domain. The coefficients of $mathcal{A}^varepsilon$ 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 $widehat{mathcal{A}}^varepsilon$, we show that the point spectrum of $mathcal{A}^epsilon$ converges in the sense of Hausdorff to the point spectrum of the homogenised operator $mathcal{A}^mathrm{hom}$ as $varepsilon to 0$. Furthermore, we prove that the eigenfunctions of $mathcal{A}^varepsilon$ decay exponentially at infinity uniformly for sufficiently small $varepsilon$. This, in turn, yields strong stochastic two-scale convergence of such eigenfunctions to eigenfunctions of $mathcal{A}^mathrm{hom}$.