In this talk we will discuss ensembles of finite dimensional spectral triples as models of Euclidean Quantum Gravity originally proposed by J. W. Barrett. Such models are matrix integrals referred to as Dirac ensembles. In recent joint work with H. Hessam and M. Khalkhali we established a connection between Dirac Ensembles and 2D Liouville Quantum Gravity. Using techniques of random matrix theory, we computed the critical exponents and the asymptotic expansion of partition functions of various Dirac ensembles which match that of minimal models from Liouville conformal field theory coupled with gravity. Further work has found explicit solutions and the critical exponents of more complicated previously unsolved Dirac ensembles.

Bio: Nathan Pagliaroli is a PhD student at Western University who studies random matrices and Noncommutative Geometry.