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Thesis Format

Integrated Article


Doctor of Philosophy




Khalkhali, Masoud.

2nd Supervisor

Barron, Tatyana.

Joint Supervisor


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.

Summary for Lay Audience

The nature of a theory of spacetime in quantum gravity is constrained by the existence of the Planck length. In fact, using the Heisenberg uncertainty principle and Einstein’s general relativity theory, it can be shown that spacetime cannot be a smooth manifold at Planck length since black holes can emerge in very small scales. Several options have been suggested to replace the conventional spacetime. One such suggestion is a noncommutative Riemannian manifold in the sense of spectral triples. In particular, for finite dimensional spectral triples, the role of the metric is played by a Dirac operator.

A random matrix is a matrix whose entries are random variables. The main goal of this study is to find the probability distribution function of eigenvalues of certain random matrices that appear in some toy models of quantum gravity based on noncommutative geometry.

In this thesis, in order to find the moments of random Dirac operator numerically, we use the bootstrapping method. The bootstrapping method is based on a set of recursive relations called “Schwinger-Dyson equations", and some positivity constraints that are satisfied by the moments of eigenvalue distributions of such matrices. We are able to find higher moments of the model in terms of the second moment, and we calculate the second moment numerically using the bootstrapping method.

In order to solve matrix ensembles when the size of the matrix reaches infinity, we offer a brand-new technique called the moment-coefficient method. This method is compatible with several well-known methods for solving single matrix ensembles. In particular, it is used to analyze Dirac ensembles at the double scaling limit, which occurs when the model's order parameter approaches the critical value and the size of the matrix reaches infinity. We show that the so called free energy of our models (logarithm of the partition function) can be constructed using solutions of the Painlevé I differential equation. Additionally, Monte Carlo Metropolis-Hastings simulations are used to numerically support the outcomes of this thesis.

Creative Commons License

Creative Commons Attribution 4.0 License
This work is licensed under a Creative Commons Attribution 4.0 License.