  The supermassive black holes (SMBHs) massive enough to power the bright
redshift ~6 quasars observed in the Sloan Digital Sky Survey (SDSS) are thought
to have assembled by mergers and/or gas accretion from less massive "seed" BHs.
If the seeds are the ~100 Msol remnant BHs from the first generation of stars,
they must be in place well before redshift z=6, and must avoid being ejected
from their parent proto-galaxies by the large (few hundred km/s) kicks they
suffer from gravitational-radiation induced recoil during mergers with other
BHs. We simulate the SMBH mass function at redshift z=6 using dark matter (DM)
halo merger trees, coupled with a prescription for the halo occupation
fraction, accretion histories, and radial recoil trajectories of the growing
BHs. Our purpose is (i) to map out plausible scenarios for successful assembly
of the z~6 quasar BHs by exploring a wide region of parameter space, and (ii)
to predict the rate of low-frequency gravitational wave events detectable by
the Laser Interferometer Space Antenna (LISA) for each such scenario. Our main
findings are as follows: (1) ~100 Msol seed BHs can grow into the SDSS quasar
BHs without super--Eddington accretion, but only if they form in minihalos by
z~30 and subsequently accrete ~60% of the time; (2) the scenarios with
optimistic assumptions required to explain the SDSS quasar BHs overproduce the
mass density in lower--mass (10^5 to 10^7 Msol) SMBHs by a factor of 100-1000,
unless seeds stop forming, or accrete at a severely diminished rates or duty
cycles (e.g. due to feedback), at z < 20-30. We also present several successful
assembly models and their LISA detection rates, including a "maximal" model
that gives the highest rate (~30/yr at z~6) without overproducing the total
SMBH density.
