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We show that the superconformal index of N=1 superconformal field theories infour dimensions has an asymptotic growth of states which is exponential in thecharges. Our analysis holds in a Cardy-like limit of large charges, for whichthe index is dominated by small values of chemical potentials. In this limit wefind the saddle points of the integral that defines the superconformal indexusing two different methods. One method, valid for finite N, is to first takethe Cardy-like limit and then find the saddle points. The other method is toanalyze the saddle points at large N and then take the Cardy-like limit. Theresult of both analyses is that the asymptotic growth of states of thesuperconformal index exactly agrees with the Bekenstein-Hawking entropy ofsupersymmetric black holes in the dual AdS$_5$ theory.

The asymptotic growth of states of the 4d N=1 superconformal index