Large $N$ Universality of 4d $\mathcal{N}=1$ Superconformal Index and AdS Black Holes

We study the large $N$ limit of the matrix models associated with the superconformal indices of four-dimensional $\mathcal{N}=1$ superconformal field theories. We find that for a large class of $\mathcal{N}=1$ superconformal gauge theories, the superconformal indices in the large $N$ limit of such theories are dominated by the 'parallelogram' saddle, providing $O(N^2)$ free energy for the generic value of chemical potentials. This saddle corresponds to BPS black holes in AdS$_5$ whenever a holographic dual description is available. Our saddle applies to a large class of gauge theories, including ADE quiver gauge theories, and the theories with rank-2 tensor matters. Our analysis works for most $\mathcal{N}=1$ superconformal gauge theories that admit a suitable large $N$ limit while keeping the flavor symmetry fixed. We also find 'multi-cut' saddle points, which correspond to the orbifolded Euclidean black holes in AdS$_5$.