We present an analytic proof of the existence of phase transition in the large N limit of certain random noncommutative geometries. These geometries can be expressed as ensembles of Dirac operators. When they reduce to single matrix ensembles, one can apply the Coulomb gas method to find the empirical spectral distribution. We elaborate on the nature of the large N spectral distribution of the Dirac operator itself. Furthermore, we show that these models exhibit both a single and double cut region for certain values of the order parameter and find the exact value where the transition occurs.

中文翻译：

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随机非交换几何中的相变

我们提供了在某些随机非交换几何的大N极限中存在相变的分析证明。这些几何形状可以表示为Dirac算子的集合。当它们简化为单个矩阵集合时，可以应用库仑气体方法来找到经验光谱分布。我们详细介绍了狄拉克算子本身的大N光谱分布的性质。此外，我们显示出这些模型对订单参数的某些值同时显示了单切和双切区域，并找到了发生过渡的确切值。