  Quantum black holes are described by a large number of macroscopically
indistinguishable microstates. Correlation functions of fields outside the
horizon at long time separation probe this indistinguishability. The simplest
of these, the thermal two-point function, oscillates erratically around a
nonperturbatively small average "ramp" and "plateau" after an initial period of
decay; these non-decaying averaged features are signatures of the discreteness
of the black hole spectrum. For a theory described by an ensemble of
Hamiltonians, the two-point function follows this averaged behavior. In this
paper we study certain correlation functions in Jackiw-Teitelboim (JT) gravity
and find precise agreement with the behavior expected for a theory described by
an ensemble of Hamiltonians with random matrix statistics -- the eigenstates
obey the Eigenstate Thermalization Hypothesis (ETH) and the energy levels have
random matrix level statistics. A central aspect of our analysis is an averaged
bulk Hilbert space description of the relevant behavior. The mechanism behind
this behavior is topology change due the the emission and absorption of closed
"baby universes". These baby universe effects give two complementary pictures
of the non-decaying behavior, related by different continuations of a Euclidean
geometry. A long Einstein-Rosen bridge can become short by emitting a large
baby universe, and baby universes emitted and reabsorbed at points widely
separated in space and time creates a "shortcut", allowing particles to leave
the interior of the black hole.
