PhD Thesis Defence Public Lecture (Math) - Hamed Hessam

Dirac ensembles are finite dimensional real spectral triples where the Dirac operator is allowed to vary within a suitable family of operators and is assumed to be random. The Dirac operator plays the role of a metric on a manifold in the noncommutative geometry context of spectral triples. Thus, integration over the set of Dirac operators within a Dirac ensemble, a crucial aspect of a theory of quantum gravity, is a noncommutative analog of integration over metrics.

Dirac ensembles are closely related to random matrix ensembles. In order to determine properties of specific Dirac ensembles, we use techniques from random matrix theory such as Schwinger-Dyson equations and the recently introduced bootstrapping. In particular, we determine the relations between the second moments of our models and parameters of the models. All the other moments can be represented in terms of the coupling constants and the second moments using the set of recursive relations called the Schwinger-Dyson equations. Additionally, explicit relations for higher mixed moments are found.

We also introduce a new technique, the moment-coefficient method, to solve multi-trace matrix models in the large $N$ limit. This technique is compatible with several well-known approaches to solving single matrix ensembles. Using this technique, we study Dirac ensembles in the so called "double scaling limit". It is significant to note that, as predicted by conformal field theory, the asymptotics of the partition function of these models is used to construct a solution for the Painlevé I differential equation. Moreover, results of this thesis are also justified numerically by Monte Carlo Metropolis-Hastings simulations.