  The presence of multiple luminous galaxies in clusters can be explained by
the finite time over which a galaxy sinks to the center of the cluster and
merges with the the central galaxy. The simplest measurable statistic to
quantify the dynamical age of a system of galaxies is the luminosity
(magnitude) gap, which is the difference in photometric magnitude between the
two most luminous galaxies. We present a simple analytical estimate of the
luminosity gap distribution in groups and clusters as a function of dark matter
halo mass. The luminosity gap is used to define "fossil" groups; we expect the
fraction of fossil systems to exhibit a strong and model-independent trend with
mass: ~1-3% of massive clusters and ~5-40% of groups should be fossil systems.
We also show that, on cluster scales, the observed intrinsic scatter in the
central galaxy luminosity-halo mass relation can be ascribed to dispersion in
the merger histories of satellites within the cluster. We compare our
predictions to the luminosity gap distribution in a sample of 730 clusters in
the Sloan Digital Sky Survey C4 Catalog and find good agreement. This suggests
that theoretical excursion set merger probabilities and the standard theory of
dynamical segregation are valid on cluster scales.
