  General Relativity simplifies dramatically in the limit that the number of
spacetime dimensions D is infinite: it reduces to a theory of non-interacting
particles, of finite radius but vanishingly small cross sections, which do not
emit nor absorb radiation of any finite frequency. Non-trivial black hole
dynamics occurs at length scales that are 1/D times smaller than the horizon
radius, and at frequencies D times larger than the inverse of this radius. This
separation of scales at large D, which is due to the large gradient of the
gravitational potential near the horizon, allows an effective theory of black
hole dynamics. We develop to leading order in 1/D this effective description
for massless scalar fields and compute analytically the scalar absorption
probability. We solve to next-to-next-to-leading order the black brane
instability, with very accurate results that improve on previous approximations
with other methods. These examples demonstrate that problems that can be
formulated in an arbitrary number of dimensions may be tractable in analytic
form, and very efficiently so, in the large D expansion.
