Large N limit of fuzzy geometries coupled to fermions.

This article analyzes the large N limit of fermionic quartic Dirac ensembles based on (0, 1)-fuzzy geometries, which are single-matrix, multi-trace Hermitian matrix ensembles incorporating fermionic contributions. The authors prove the existence and uniqueness of the large N spectral density (equilibrium measure) for these multi-tracial, non-polynomial matrix models and develop spectral estimators—specifically spectral dimension and spectral variance—to extract geometric information from the normalized heat kernel, as the usual heat kernel techniques are inapplicable due to bounded spectra. They investigate the effects of adding a fermionic action with mass terms on the spectral density, phase transitions between one-cut and two-cut spectral phases, and spectral estimators, showing that fermionic contributions modify eigenvalue repulsion and coupling constants, thereby influencing the geometry encoded by the Dirac operator. The paper also discusses extensions to multiple fermions and connections to random matrix theory and noncommutative geometry, providing integral equations for equilibrium measures and numerical analyses of spectral properties under varying model parameters.