RedMaPPer: Evolution and Mass Dependence of the Conditional Luminosity
  Functions of Red Galaxies in Galaxy Clusters
  We characterize the luminosity distribution, halo mass dependence, and
redshift evolution of red galaxies in galaxy clusters using the SDSS Data
Release 8 RedMaPPer cluster sample. We propose a simple prescription for the
relationship between the luminosity of both central and satellite galaxies and
the mass of their host halos, and show that this model is well-fit by the data.
Using a larger galaxy cluster sample than previously employed in the
literature, we find that the luminosities of central galaxies scale as $\langle
\log L \rangle \propto A_L \log (M_{200b})$, with $A_L=0.39\pm0.04$, and that
the scatter of the central--galaxy luminosity at fixed $M_{200b}$ (
$\sigma_{\log L|M}$) is $0.23 ^{+0.05}_{-0.04}$ dex, with the error bar
including systematics due to miscentering of the cluster finder, photometry,
and photometric redshift estimation. Our data prefers a positive correlation
between the luminosity of central galaxies and the observed richness of
clusters at a fixed halo mass, with an effective correlation coefficient
$d_{\rm{eff}}=0.36^{+0.17}_{-0.16}$. The characteristic luminosity of
satellites becomes dimmer from $z=0.3$ to $z=0.1$ by $\sim 20\%$ after
accounting for passive evolution. We estimate the fraction of galaxy clusters
where the brightest galaxy is not the central to be $P_{\rm{BNC}} \sim 20\%$.
We discuss implications of these findings in the context of galaxy evolution
and the galaxy--halo connection.
