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Modular covariance of torus one-point functions constrains the three pointfunction coefficients of a two dimensional CFT. This leads to an asymptoticformula for the average value of light-heavy-heavy three point coefficients,generalizing Cardy's formula for the high energy density of states. Thederivation uses certain asymptotic properties of one-point conformal blocks onthe torus. Our asymptotic formula matches a dual AdS_3 computation of one pointfunctions in a black hole background. This is evidence that the BTZ black holegeometry emerges upon course-graining over a suitable family of heavymicrostates.

A Cardy Formula for Three-Point Coefficients: How the Black Hole Got its Spots