  We study the orthogonality catastrophe due to a parametric change of the
single-particle (mean field) Hamiltonian of an ergodic system. The Hamiltonian
is modeled by a suitable random matrix ensemble. We show that the overlap
between the original and the parametrically modified many-body ground states,
$S$, taken as Slater determinants, decreases like $n^{-k x^2}$, where $n$ is
the number of electrons in the systems, $k$ is a numerical constant of the
order of one, and $x$ is the deformation measured in units of the typical
distance between anticrossings. We show that the statistical fluctuations of
$S$ are largely due to properties of the levels near the Fermi energy.
