Numerical simulation of random Dirac operators - D'Arcangelo, Mauro (2022)
A Euclidean path integral over matrix Dirac operators associated to fuzzy spaces is investigated using analytical and numerical tools of random matrix theory. A numerical library for handling Monte Carlo integration of fuzzy Dirac operators is written and tested. The random matrix theory arising from the simplest class of fuzzy Dirac operators is solved exactly using the theory of Riemann-Hilbert problems, and the results are confirmed numerically. For higher classes of Dirac operators, where integration is extended over many Hermitian matrices, various local minima of the action are found by solving the equations of motion. Among others, su(2) solutions are shown to exist, and strong evidence is given of their realization in the asymptotic regime of the random model. Numerical data is collected in the vicinity of phase transitions occurring in various models, and it is shown how in certain cases they can be interpreted as transitions between a commutative and a non-commutative regime. Finally, a link is established between the action of fuzzy Dirac operators and Yang-Mills matrix models.