  A two-dimensional CFT dual to a semiclassical theory of gravity in three
dimensions must have a large central charge $c$ and a sparse low energy
spectrum. This constrains the OPE coefficients and density of states of the CFT
via the conformal bootstrap. We define an ensemble of CFT data by averaging
over OPE coefficients subject to these bootstrap constraints, and show that
calculations in this ensemble reproduce semiclassical 3D gravity. We analyze a
wide variety of gravitational solutions, both in pure Einstein gravity and
gravity coupled to massive point particles, including Euclidean wormholes with
multiple boundaries and higher topology spacetimes with a single boundary. In
all cases we find that the on-shell action of gravity agrees with the
ensemble-averaged CFT at large $c$. The one-loop corrections also match in the
cases where they have been computed. We also show that the bulk effective
theory has random couplings induced by wormholes, providing a controlled,
semiclassical realization of the mechanism of Coleman, Giddings, and
Strominger.
