  We study the evolution of black holes (BHs) on the M_BH-sigma and
M_BH-M_bulge planes as a function of time in disk galaxies undergoing mergers.
We begin the simulations with the progenitor black hole masses being initially
below (Delta log M_BH=-2), on (Delta log M_BH=0) and above (Delta log M_BH=0.5)
the observed local relations. The final relations are rapidly established after
the final coalescense of the galaxies and their BHs. Progenitors with low
initial gas fractions (f_gas=0.2) starting below the relations evolve onto the
relations (Delta log M_BH=-0.18), progenitors on the relations stay there
(Delta log M_BH=0) and finally progenitors above the relations evolve towards
the relations, but still remaining above them (Delta log M_BH=0.35). Mergers in
which the progenitors have high initial gas fractions (f_gas=0.8) evolve above
the relations in all cases (Delta log M_BH=0.5). We find that the initial gas
fraction is the prime source of scatter in the observed relations, dominating
over the scatter arising from the evolutionary stage of the merger remnants.
The fact that BHs starting above the relations do not evolve onto the
relations, indicates that our simulations rule out the scenario in which
overmassive BHs evolve onto the relations through gas-rich mergers. By
implication our simulations thus disfavor the picture in which supermassive BHs
develop significantly before their parent bulges.
