Universal asymptotics of three-point coefficients from elliptic representation of Virasoro blocks

In (1+1)-d CFTs, the 4-point function on the plane can be mapped to the pillow geometry and thereby crossing symmetry gets translated into a modular property. This feature arises from the Virasoro blocks in the elliptic representation. We use these modular features to derive a universal asymptotic formula for OPE coefficients in which one of the operators is averaged over heavy primaries. As an application, we demonstrate that the coarse-grained heavy channel then reproduces features of the holographic 2→2 S-matrix which has black holes as their intermediate states.