  In this paper we study ensembles of finite real spectral triples equipped
with a path integral over the space of possible Dirac operators. In the
noncommutative geometric setting of spectral triples, Dirac operators take the
center stage as a replacement for a metric on a manifold. Thus, this path
integral serves as a noncommutative analogue of integration over metrics, a key
feature of a theory of quantum gravity. From these integrals in the so-called
double scaling limit we derive critical exponents of minimal models from
Liouville conformal field theory coupled with gravity. Additionally, the
asymptotics of the partition function of these models satisfy differential
equations such as Painlev\'e I, as a reduction of the KDV hierarchy, which is
predicted by conformal field theory. This is all proven using well-established
and rigorous techniques from random matrix theory.
