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We present an analytic proof of the existence of phase transition in thelarge $N$ limit of certain random noncommutaitve geometries. These geometriescan be expressed as ensembles of Dirac operators. When they reduce to singlematrix ensembles, one can apply the Coulomb gas method to find the empiricalspectral distribution. We elaborate on the nature of the large $N$ spectraldistribution of the Dirac operator itself. Furthermore, we show that thesemodels exhibit both a single and double cut region for certain values of theorder parameter and find the exact value where the transition occurs.

Phase Transition in Random Noncommutative Geometries