  Two-dimensional conformal field theories at large central charge and with a
sufficiently sparse spectrum of light states have been shown to exhibit
universal thermodynamics. This thermodynamics matches that of AdS$_3$ gravity,
with a Hawking-Page transition between thermal AdS and the BTZ black hole. We
extend these results to correlation functions of light operators. Upon making
some additional assumptions, such as large $c$ factorization of correlators, we
establish that the thermal AdS and BTZ solutions emerge as the universal
backgrounds for the computation of correlators. In particular, Witten diagrams
computed on these backgrounds yield the CFT correlators, order by order in a
large $c$ expansion, with exponentially small corrections. In pure CFT terms,
our result is that thermal correlators of light operators are determined
entirely by light spectrum data. Our analysis is based on the constraints of
modular invariance applied to the torus two-point function.
