  We investigate the statistical orthogonality catastrophe (StOC) in
single-particle and many-body localized systems by studying the response of the
many-body ground state to a local quench. Using scaling arguments and exact
numerical calculations, we establish that the StOC gives rise to a wave
function overlap between the pre- and post-quench ground states that has an
\emph{exponential} decay with the system size, in sharp contrast to the
well-known power law Anderson orthogonality catastrophe in metallic systems.
This exponential decay arises from a statistical charge transfer process where
a particle can be effectively "transported" to an arbitrary lattice site. In a
many-body localized phase, this non-local transport and the associated
exponential StOC phenomenon persist in the presence of interactions. We study
experimental consequences of the exponential StOC on Loschmidt echo and
spectral function, establishing that this phenomenon should be observable in
cold atomic experiments through Ramsey interference and radio-frequency
spectroscopy.
