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 $N\to \mathrm{\infty }$ 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.