  We use the high-resolution cosmological simulation Illustris to investigate
the clustering of supermassive black holes across cosmic time, the link between
black hole clustering and host halo masses, and the implications for black hole
duty cycles. Our predicted black hole correlation length and bias match the
observational data very well across the full redshift range probed. Black hole
clustering is strongly luminosity-dependent on small, 1-halo scales, with some
moderate dependence on larger scales of a few Mpc at intermediate redshifts. We
find black hole clustering to evolve only weakly with redshift, initially
following the behaviour of their hosts. However below z ~ 2 black hole
clustering increases faster than that of their hosts, which leads to a
significant overestimate of the clustering-predicted host halo mass. The full
distribution of host halo masses is very wide, including a low-mass tail
extending up to an order of magnitude below the naive prediction for minimum
host mass. Our black hole duty cycles follow a power-law dependence on black
hole mass and decrease with redshift, and we provide accurate analytic fits to
these. The increase in clustering amplitude at late times, however, means that
duty cycle estimates based on black hole clustering can overestimate duty
cycles substantially, by more than two orders of magnitude. We find the best
agreement when the minimum host mass is assumed to be $10^{11.2} M_\odot$ ,
which provides an accurate measure across all redshifts and luminosity ranges
probed by our simulation.
