  Recent observations from the EIGER JWST program have measured for the first
time the quasar-galaxy cross-correlation function at $z\approx6$. The
auto-correlation function of faint $z\approx6$ quasars was also recently
estimated. These measurements provide key insights into the properties of
quasars and galaxies at high redshift and their relation with the host dark
matter halos. In this work, we interpret these data building upon an empirical
quasar population model that has been applied successfully to quasar clustering
and demographic measurements at $z\approx2-4$. We make use of a new,
large-volume N-body simulation with more than a trillion particles,
FLAMINGO-10k, to model quasars and galaxies simultaneously. We successfully
reproduce observations of $z\approx6$ quasars and galaxies (i.e., their
clustering properties and luminosity functions), and infer key quantities such
as their luminosity-halo mass relation, the mass function of their host halos,
and their duty cycle/occupation fraction. Our key findings are: (i) quasars
reside on average in $\approx10^{12.4}\,{\rm M}_\odot$ halos (corresponding to
$\approx5\sigma$ fluctuations in the initial conditions of the linear density
field), but the distribution of host halo masses is quite broad; (ii) the duty
cycle of (UV-bright) quasar activity is relatively low ($\lesssim1\%$); (iii)
galaxies (that are bright in [OIII]) live in much smaller halos
($\approx10^{10.9}\,{\rm M}_\odot$) and have a larger duty cycle (occupation
fraction) of $\approx20\%$. Finally, we focus on the inferred properties of
quasars and present a homogeneous analysis of their evolution with redshift.
The picture that emerges reveals a strong evolution of the host halo mass and
duty cycle of quasars at $z\approx2-6$, and calls for new investigations of the
role of quasar activity across cosmic time.
