LAUSR

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.