A periodic homogenization problem with defects rare at infinity

We consider a homogenization problem for the diffusion equation $-\mathrm{div}\left(a_{\epsilon }\mathrm{\nabla }{u}_{\epsilon }\right)=f$ when the coefficient $a_{\epsilon }$ is a non-local perturbation of a periodic coefficient. The perturbation does not vanish but becomes rare at infinity in a sense made precise in the text. We prove the existence of a corrector, identify the homogenized limit and study the convergence rates of $u_{\epsilon }$ to its homogenized limit.