From Noncommutative Geometry to Random Matrix Theory

Abstract

We review recent progress in the analytic study of random matrix models suggested by noncommutative geometry. One considers finite real spectral triples where the space of possible Dirac operators is assigned a probability distribution. These Dirac ensembles are constructed as toy models of Euclidean quantum gravity on finite noncommutative spaces and display many interesting properties. Near phase transitions they exhibit manifold like behavior and 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.