  We discovered a significant anti-correlation between the mass of a
supermassive black hole (SMBH), $M_{\rm BH}$, and the luminosity ratio of
infrared to active galactic nuclei (AGN) Eddington luminosity, $L_{\rm
IR}/L_{\rm Edd}$, over four orders of magnitude for ultraluminous infrared
galaxies with type I Seyfert nuclei (type I ULIRGs) and nearby QSOs. This
anti-correlation ($M_{\rm BH}$ vs. $L_{\rm IR}/L_{\rm Edd}$) can be interpreted
as the anti-correlation between the mass of a SMBH and the rate of mass
accretion onto a SMBH normalized by the AGN Eddington rate, $\dot{M}_{\rm
BH}/\dot{M}_{\rm Edd}$. In other words, the mass accretion rate $\dot{M}_{\rm
BH}$ is not proportional to that of the central BH mass. Thus, this
anti-correlation indicates that BH growth is determined by the external mass
supply process, and not the AGN Eddington-limited mechanism. Moreover, we found
an interesting tendency for type I ULIRGs to favor a super-Eddington accretion
flow, whereas QSOs tended to show a sub-Eddington flow. On the basis of our
findings, we suggest that a central SMBH grows by changing its mass accretion
rate from super-Eddington to sub-Eddington. According to a coevolution scenario
of ULIRGs and QSOs based on the radiation drag process, it has been predicted
that a self-gravitating massive torus, whose mass is larger than a central BH,
exists in the early phase of BH growth (type I ULIRG phase) but not in the
final phase of BH growth (QSO phase). At the same time, if one considers the
mass accretion rate onto a central SMBH via a turbulent viscosity, the
anti-correlation ($M_{\rm BH}$ vs. $L_{\rm IR}/L_{\rm Edd}$) is well explained
by the positive correlation between the mass accretion rate $\dot{M}_{\rm BH}$
and the mass ratio of a massive torus to a SMBH.
