  The study reports on the first large statistics numerical experiment
searching for rare eigenstates of anomalously high amplitudes in
three-dimensional diffusive metallic conductors. Only a small fraction of a
huge number of investigated eigenfunctions generates the far asymptotic tail of
their amplitude distribution function. The relevance of the relationship
between disorder and spectral averaging, as well as of the quantum transport
properties of the investigated mesoscopic samples, for the numerical
exploration of eigenstate statistics is divulged. The quest provides exact
results to serve as a reference point in understanding the limits of
approximations employed in different analytical predictions, and thereby the
physics (quantum vs semiclassical) behind large deviations from the universal
predictions of random matrix theory.
