Symmetry Breaking and Phase Transitions in Random Non-Commutative Geometries and Related Random-Matrix Ensembles

Ensembles of random fuzzy non-commutative geometries may be described in terms of finite (\(N^2\)-dimensional) Dirac operators and a probability measure. Dirac operators of type \((p,q)\) are defined in terms of commutators and anti-commutators of \(2^{p+q-1}\) hermitian matrices \(H_k\) and tensor products with a representation of a Clifford algebra. Ensembles based on this idea have recently been used as a toy model for quantum gravity, and they are interesting random-matrix ensembles in their own right. We provide a complete theoretical picture of crossovers, phase transitions, and symmetry breaking in the \(N \to \infty \) limit of 1-parameter families of quartic Barrett-Glaser ensembles in the one-matrix cases \((1,0)\) and \((0,1)\) that depend on one coupling constant \(g\). Our theoretical results are in full agreement with previous and new Monte-Carlo simulations.