A periodic homogenization problem with defects rare at infinity

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