The Evolution of K* and the Halo Occupation Distribution since z=1.5:
  Observations vs. Simulations
  We study the evolution of the K-band luminosity function (LF) and the Halo
Occupation Distribution (HOD) using Subaru observations of 15 X-ray clusters at
z=0.8-1.5 and compare the results with mock clusters (0<z<1.3) extracted from
the Millennium Simulation and populated with galaxies using the semi-analytic
model (SAM) of Bower et al., matched in mass to our observed sample. We find
that the characteristic luminosity K* defined by a Shechter LF is consistent
with SAM predictions, which mimic well the evolution of K* in z>1 rich
clusters. However, we cannot distinguish between this model and a simple
stellar population synthesis model invoking passive evolution with a formation
redshift z~5 - consistent with the presence of an old red galaxy population
ubiquitous in rich clusters at z=1.5. We also see a small difference (\Delta
K*~0.5) between our clusters and studies of the field population at similar
redshifts, suggesting only a weak dependence of the luminous (L>L*) part of the
LF on environment. Turning to our HOD study, we find that within R_{500},
high-z clusters tend to host smaller numbers of galaxies to a magnitude K*+2
compared to their low-z counterparts. This behavior is also seen in the mock
samples and is relatively insensitive to the average mass of the cluster
haloes. In particular, we find significant correlations of the observed number
of member cluster galaxies (N) with both z and cluster mass:
$N(M,z)=(53\pm1)(1+z)^{-0.61^{+0.18}_{-0.20}}(M/10^{14.3})^{0.86\pm0.05}$.
Finally, we examine the spatial distribution of galaxies and provide a new
estimate of the concentration parameter for clusters at high z
($c_{g}=2.8^{+1.0}_{-0.8}$). Our result is consistent with predictions from
both our SAM mock clusters and literature's predictions for dark matter haloes.
The mock sample predictions rise slowly with decreasing redshift reaching
$c_{g}=6.3^{+0.39}_{-0.36}$ at z=0.
