In this paper, we consider Anderson type operators on a separable Hilbert space where the random perturbations are finite rank and the random variables have full support on [Formula: see text]. We show that spectral multiplicity has a uniform lower bound whenever the lower bound is given on a set of positive Lebesgue measure on the point spectrum away from the continuous one. We also show a deep connection between the multiplicity of pure point spectrum and local spectral statistics, in particular, we show that spectral multiplicity higher than one always gives non-Poisson local statistics in the framework of Minami theory.In particular, for higher rank Anderson models with pure point spectrum, with the randomness having support equal to [Formula: see text], there is a uniform lower bound on spectral multiplicity and in case this is larger than one, the local statistics is not Poisson.

中文翻译：

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随机算子的全局多重性界限和谱统计

在本文中，我们考虑可分离希尔伯特空间上的安德森类型算子，其中随机扰动是有限秩的，并且随机变量完全支持 [公式：见文本]。我们表明，只要在远离连续谱的点谱上的一组正 Lebesgue 测度上给出下限，谱多重性就有一个统一的下限。我们还展示了纯点谱的多重性与局部谱统计之间的深层联系，特别是，我们表明在 Minami 理论的框架下，谱多重性高于 1 总是给出非泊松局部统计。具有纯点谱的模型，具有支持等于 [公式：见文本] 的随机性，