  We review recent progress in the analytic study of random matrix models
suggested by noncommutative geometry. One considers fuzzy spectral triples
where the space of possible Dirac operators is assigned a probability
distribution. These ensembles of Dirac operators are constructed as toy models
of Euclidean quantum gravity on finite noncommutative spaces and display many
interesting properties. The ensembles exhibit spectral phase transitions, and
near these phase transitions they show manifold-like behavior. In certain cases
one can recover Liouville quantum gravity in the double scaling limit. We
highlight examples where bootstrap techniques, Coulomb gas methods, and
Topological Recursion are applicable.
