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 are random, possess double porosity type scaling, and are perturbed on a fixed-size compact domain (a defect). Working in the gaps of the limiting spectrum of the unperturbed operator A widehat ϵ, we show that the point spectrum of A ϵ converges in the sense of Hausdorff to the point spectrum of the limiting two-scale operator A h o m 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 h o m .