  We address the phenomenon of statistical orthogonality catastrophe in
insulating disordered systems. More in detail, we analyse the response of a
system of non-interacting fermions to a local perturbation induced by an
impurity. By inspecting the overlap between the pre and post-quench many-body
ground states we fully characterise the emergent statistics of orthogonality
events as a function of both the impurity position and the coupling strength.
We consider two well-known one-dimensional models, namely the Anderson and the
Aubry- Andr\'e insulators, highlighting the arising differences. Particularly,
in the Aubry-Andr\'e model the highly correlated nature of the quasi periodic
potential produces unexpected features in how the orthogonality catastrophe
occurs. We provide a quantitative explanation of such features via a simple,
effective model. We further discuss the incommensurate ratio approximation and
suggest a viable experimental verification in terms of charge transfer
statistics and interferometric experiments using quantum probes.
