Universality in the OPE Coefficients of Holographic 2d CFTs

The thermodynamic stability of large AdS$_3$ black holes implies that Cardy's $\Delta\rightarrow\infty$ formula for the density of states remains approximately valid when $\Delta\sim c$ in holographic 2d CFTs, constraining their light spectra. Averaged OPE coefficients take a similarly universal asymptotic form, and black hole arguments again imply an extended regime of validity. In this note we study conditions under which the OPE asymptotics extend to $\Delta\sim c$ at large central charge. Some of the conditions found are stronger than required by an extended Cardy regime and are violated by permutation orbifolds, such as the D1-D5 system at zero coupling. Our results suggest new bounds on non-vacuum block contributions to correlation functions in holographic CFTs.