The mass assembly of high-redshift black holes
  We use the Delphi semi-analytic model to study the mass assembly and
properties of high-redshift ($z>4$) black holes over a wide mass range, $10^3 <
M_{bh}/M_\odot < 10^{10}$. Our black hole growth implementation includes a
critical halo mass ($M_{h}^{crit}$) below which the black hole is starved and
above which it is allowed to grow either at the Eddington limit or
proportionally to the gas content of the galaxy. As a consequence, after an
initial growth phase dominated by black hole mergers down to $z \sim 7 (9)$,
supermassive black holes in $z=4$ halo masses of $M_h|_{z=4} \sim 10^{11.75} ~
(10^{13.4}) M_\odot$ mainly grow by gas accretion from the interstellar medium.
In particular, we find that: (i) while most of the accretion occurs in the
major branch for $M_h|_{z=4} \sim 10^{11-12} M_\odot$ halos, accretion in
secondary branches plays a significant role in assembling the black hole mass
in higher-mass halos ($M_h|_{z=4} \gtrsim 10^{12} M_\odot$); (ii) while the
Eddington ratio increases with decreasing redshift for low-mass ($M_{bh} < 10^5
M_\odot$) black holes, it shows the opposite trend for larger masses. In
addition, since the accretion rate depends on the gas mass present in the host
halo, the duty cycle of the Eddington-limited accretion phase -- which can last
up to $\approx 650$ Myr -- is crucially linked to the joint assembly history of
the black hole and its host halo.
