Magnitude Gap Statistics and the Conditional Luminosity Function
  In a recent preprint, Hearin et al. (2012,H12) suggest that the halo
mass-richness calibration of clusters can be improved by using the difference
in the magnitude of the brightest and the second brightest galaxy (magnitude
gap) as an additional observable. They claim that their results are at odds
with the results from Paranjape & Sheth (2012, PS12) who show that the
magnitude distribution of the brightest and second brightest galaxies can be
explained based on order statistics of luminosities randomly sampled from the
total galaxy luminosity function. We find that a conditional luminosity
function (CLF) for galaxies which varies with halo mass, in a manner which is
consistent with existing observations, naturally leads to a magnitude gap
distribution which changes as a function of halo mass at fixed richness, in
qualitative agreement with H12. We show that, in general, the luminosity
distribution of the brightest and the second brightest galaxy depends upon
whether the luminosities of galaxies are drawn from the CLF or the global
luminosity function. However, we also show that the difference between the two
cases is small enough to evade detection in the small sample investigated by
PS12. This shows that the luminosity distribution is not the appropriate
statistic to distinguish between the two cases, given the small sample size. We
argue in favor of the CLF (and therefore H12) based upon its consistency with
other independent observations, such as the kinematics of satellite galaxies,
the abundance and clustering of galaxies, and the galaxy-galaxy lensing signal
from the Sloan Digital Sky Survey.
