  We compute four-point functions of two heavy and two "perturbatively heavy"
operators in the semiclassical limit of Liouville theory on the sphere. We
obtain these "Heavy-Heavy-Light-Light" (HHLL) correlators to leading order in
the conformal weights of the light insertions in two ways: (a) via a path
integral approach, combining different methods to evaluate correlation
functions from complex solutions for the Liouville field, and (b) via the
conformal block expansion. This latter approach identifies an integral over the
continuum of normalizable states and a sum over an infinite tower of lighter
discrete states, whose contribution we extract by analytically continuing
standard results to our HHLL setting. The sum over this tower reproduces the
sum over those complex saddlepoints of the path integral that contribute to the
correlator. Our path integral computations reveal that when the two light
operators are inserted at equal time in radial quantization, the leading-order
HHLL correlator is independent of their separation, and more generally that at
this order there is no short-distance singularity as the two light operators
approach each other. The conformal block expansion likewise shows that in the
discrete sum short-distance singularities are indeed absent for all
intermediate states that contribute. In particular, the Virasoro vacuum block,
which would have been singular at short distances, is not exchanged. The
separation-independence of equal-time correlators is due to cancelations
between the discrete contributions. These features lead to a Lorentzian
singularity that, in conformal theories with anti-de Sitter (AdS) duals, would
be associated to locality below the AdS scale.
