A Direct Measurement of the Mean Occupation Function of Quasars:
  Breaking Degeneracies between Halo Occupation Distribution Models
  Recent work on quasar clustering suggests a degeneracy in the halo occupation
distribution constrained from two-point correlation functions. To break this
degeneracy, we make the first empirical measurement of the mean occupation
function (MOF) of quasars at $z \sim 0.2$ by matching quasar positions with
groups and clusters identified in the MaxBCG sample. We fit two models to the
MOF, a power law and a 4-parameter model. The number distribution of quasars in
host halos is close to Poisson, and the slopes of the MOF obtained from our
best-fit models (for the power law case) favor a MOF that monotonically
increases with halo mass. The best-fit slopes are $0.53 \pm 0.04$ and $1.03\pm
1.12$ for the power law model and the 4-parameter model, respectively. We
measure the radial distribution of quasars within dark matter halos and find it
to be adequately described by a power law with a slope $-2.3 \pm 0.4$. We
measure the conditional luminosity function (CLF) of quasars and show that
there is no evidence that quasar luminosity depends on host halo mass, similar
to the inferences drawn from clustering measurements. We also measure the
conditional black hole mass function (CMF) of our quasars. Although the results
are consistent with no dependence on halo mass, we observe a slight indication
of downsizing of the black hole mass function. The lack of halo mass dependence
in the CLF and CMF shows that quasars residing in galaxy clusters have
characteristic luminosity and black hole mass scales.
