  We construct large $N$ saddle points of the matrix model for the
$\mathcal{N}=4$ Yang-Mills index dual to the BPS black holes in $AdS_5\times
S^5$, in two different setups. When the two complex chemical potentials for the
angular momenta are collinear, we find linear eigenvalue distributions which
solve the large $N$ saddle point equation. When the chemical potentials are not
collinear, we find novel solutions given by areal eigenvalue distributions
after slightly reformulating the saddle point problem. We also construct a
class of multi-cut saddle points, showing that they sometimes admit nontrivial
filling fractions. As a byproduct, we find that the Bethe ansatz equation
emerges from our saddle point equation.
