PhD Thesis Defence Public Lecture (Math ) - Hamed Hessam

Dirac ensembles are finite real spectral triples equipped with a path integral over the space of possible Dirac operators. Dirac operator play the role in the noncommutative geometric context of spectral triples as a replacement for a metric on a manifold. Thus, the path integral, a crucial aspect of a theory of quantum gravity, behaves as a noncommutative analog of integration over metrics.

In particular, for fuzzy spectral triples, a subclass of finite real spectral triples, Dirac ensembles are precisely random matrix ensembles. In order to determine properties of specific Dirac ensembles, we use the bootstrap method. In particular, we determine the relations between the second moments of these models and parameters of the models. All the other moments can be represented in terms of the coupling constant and the second moment 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 Painlev'e I differential equation. Moreover, results of this thesis are also justified numerically by Monte Carlo Metropolis-Hasting simulations.