  We present a statistical study of the luminosity functions of galaxies
surrounding luminous red galaxies (LRGs) at average redshifts <z>=0.34 and
<z>=0.65. The luminosity functions are derived by extracting source photometry
around more than 40,000 LRGs and subtracting foreground and background
contamination using randomly selected control fields. We show that at both
studied redshifts the average luminosity functions of the LRGs and their
satellite galaxies are poorly fitted by a Schechter function due to a
luminosity gap between the centrals and their most luminous satellites. We
utilize a two-component fit of a Schechter function plus a log-normal
distribution to demonstrate that LRGs are typically brighter than their most
luminous satellite by roughly 1.3 magnitudes. This luminosity gap implies that
interactions within LRG environments are typically restricted to minor mergers
with mass ratios of 1:4 or lower. The luminosity functions further imply that
roughly 35% of the mass in the environment is locked in the LRG itself,
supporting the idea that mass growth through major mergers within the
environment is unlikely. Lastly, we show that the luminosity gap may be at
least partially explained by the selection of LRGs as the gap can be reproduced
by sparsely sampling a Schechter function. In that case LRGs may represent only
a small fraction of central galaxies in similar mass halos.
