On the radiative efficiencies, Eddington ratios, and duty cycles of
  luminous high-redshift quasars
  We investigate the characteristic radiative efficiency \epsilon, Eddington
ratio \lambda, and duty cycle P_0 of high-redshift active galactic nuclei
(AGN), drawing on measurements of the AGN luminosity function at z=3-6 and,
especially, on recent measurements of quasar clustering at z=3-4.5 from the
Sloan Digital Sky Survey. The free parameters of our models are \epsilon,
\lambda, and the normalization, scatter, and redshift evolution of the relation
between black hole mass \mbh and halo virial velocity V_vir. We compute the
luminosity function from the implied growth of the black hole mass function and
the quasar correlation length from the bias of the host halos. We test our
adopted formulae for the halo mass function and halo bias against measurements
from the large N-body simulation developed by the MICE collaboration. The
strong clustering of AGNs observed at z=3 and, especially, at z=4 implies that
massive black holes reside in rare, massive dark matter halos. Reproducing the
observed luminosity function then requires high efficiency \epsilon and/or low
Eddington ratio \lambda, with a lower limit (based on 2\sigma agreement with
the measured z=4 correlation length) \epsilon> 0.7\lambda/(1+0.7\lambda),
implying \epsilon > 0.17 for \lambda > 0.25. Successful models predict high
duty cycles, P_0~0.2, 0.5, and 0.9 at z=3.1, 4.5 and 6, respectively, and they
require that the fraction of halo baryons locked in the central black hole is
much larger than the locally observed value. The rapid drop in the abundance of
the massive and rare host halos at z>7 implies a proportionally rapid decline
in the number density of luminous quasars, much stronger than simple
extrapolations of the z=3-6 luminosity function would predict. (abridged)
