Stochastic Processes as the Origin of the Double-Power Law Shape of the
  Quasar Luminosity Function
  The Quasar Luminosity Function (QLF) offers insight into the early
co-evolution of black holes and galaxies. It has been characterized
observationally up to redshift $z\sim6$ with clear evidence of a double
power-law shape, in contrast to the Schechter-like form of the underlying
dark-matter halo mass function. We investigate a physical origin for the
difference in these distributions by considering the impact of stochasticity
induced by the processes that determine the quasar luminosity for a given host
halo and redshift. We employ a conditional luminosity function and construct
the relation between median quasar magnitude versus halo mass
$M_{UV,\rm{c}}(M_{\rm{h}})$ with log-normal in luminosity scatter $\Sigma$, and
duty-cycle $\epsilon_{\rm{DC}}$, and focus on high redshift $z\gtrsim4$. We
show that, in order to reproduce the observed QLF, the $\Sigma=0$ abundance
matching requires all of the brightest quasars to be hosted in the rarest most
massive dark-matter halos (with an increasing $M_{UV,\rm{c}}/M_{\rm{h}}$ in
halo mass). Conversely, for $\Sigma>0$ the brightest quasars can be
over-luminous outliers hosted in relatively common dark-matter halos. In this
case, the median quasar magnitude versus halo mass relation, $M_{UV,\rm{c}}$,
flattens at the high-end, as expected in self-regulated growth due to feedback.
We sample the parameter space of $\Sigma$ and $\epsilon_{\rm{DC}}$ and show
that $M_{UV,\rm{c}}$ flattens above $M_{\rm{h}}\sim 10^{12}M_{\odot}$ for
$\epsilon_{\rm{DC}}<10^{-2}$. Models with $\epsilon_{\rm{DC}}\sim1$ instead
require a high mass threshold close to $M_{\rm{h}}\gtrsim10^{13}M_{\odot}$. We
investigate the impact of $\epsilon_{\rm{DC}}$ and $\Sigma$ on measurements of
clustering and find there is no luminosity dependence on clustering for
$\Sigma>0.3$, consistent with recent observations from Subaru HSC.
