  [Abridged] Recent numerical relativity simulations have shown that the
emission of gravitational waves during the merger of two supermassive black
holes (SMBHs) delivers a kick to the final hole, with a magnitude as large as
4000 km/s. We study the motion of SMBHs ejected from galaxy cores by such kicks
and the effects on the stellar distribution using high-accuracy direct N-body
simulations. Following the kick, the motion of the SMBH exhibits three distinct
phases. (1) The SMBH oscillates with decreasing amplitude, losing energy via
dynamical friction each time it passes through the core. Chandrasekhar's theory
accurately reproduces the motion of the SMBH in this regime if 2 < ln Lambda <
3 and if the changing core density is taken into account. (2) When the
amplitude of the motion has fallen to roughly the core radius, the SMBH and
core begin to exhibit oscillations about their common center of mass. These
oscillations decay with a time constant that is at least 10 times longer than
would be predicted by naive application of the dynamical friction formula. (3)
Eventually, the SMBH reaches thermal equilibrium with the stars. We estimate
the time for the SMBH's oscillations to damp to the Brownian level in real
galaxies and infer times as long as 1 Gyr in the brightest galaxies. Ejection
of SMBHs also results in a lowered density of stars near the galaxy center;
mass deficits as large as five times the SMBH mass are produced for kick
velocities near the escape velocity. We compare the N-body density profiles
with luminosity profiles of early-type galaxies in Virgo and show that even the
largest observed cores can be reproduced by the kicks, without the need to
postulate hypermassive binary SMBHs. Implications for displaced AGNs and
helical radio structures are discussed.
