The generalized Rosenzweig--Porter model arguably constitutes the simplest random matrix ensemble displaying a non-ergodic delocalized phase, which we characterize here by using replica methods. We first derive analytical expressions for the average spectral density in the limit in which the size $N$ of the matrix is large but finite. We then focus on the number of eigenvalues in a finite interval and compute its cumulant generating function as well as the level compressibility, i.e., the ratio of the first two cumulants: these are useful tools to describe the local level statistics. In particular, the level compressibility is shown to be described by a universal scaling function, which we compute explicitly, when the system is probed over scales of the order of the Thouless energy. We confirm our results with numerical tests.

中文翻译：

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广义 Rosenzweig-Porter 模型的复制方法

广义 Rosenzweig--Porter 模型可以说是最简单的随机矩阵集合，显示出非遍历离域相位，我们在这里使用复制方法对其进行表征。我们首先在矩阵的大小 $N$ 很大但有限的情况下推导出平均谱密度的解析表达式。然后，我们关注有限区间内的特征值的数量，并计算其累积量生成函数以及级别可压缩性，即前两个累积量的比率：这些是描述局部级别统计数据的有用工具。特别是，当系统在 Thouless 能量量级的尺度上进行探测时，我们明确计算了通用尺度函数来描述能级可压缩性。我们通过数值测试证实了我们的结果。