We prove a sharp $\sqrt{N}$ transition for the infinitesimal distribution of a periodically banded GUE matrix. For band widths $b_N = \Omega(\sqrt{N})$, we further prove that our model is infinitesimally free from the matrix units and the normalized all-ones matrix. Our results allow us to extend previous work of Shlyakhtenko on finite-rank perturbations of Wigner matrices in the infinitesimal framework. For finite-rank perturbations of our model, we find outliers at the classical positions from the deformed Wigner ensemble.

中文翻译：

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基于无穷小自由概率的随机带矩阵的有限秩扰动

我们证明了周期性带状 GUE 矩阵的无穷小分布的急剧 $\sqrt{N}$ 转换。对于带宽 $b_N = \Omega(\sqrt{N})$，我们进一步证明我们的模型无限小地摆脱了矩阵单位和归一化的全一矩阵。我们的结果使我们能够扩展 Shlyakhtenko 之前在无穷小框架中 Wigner 矩阵的有限秩扰动方面的工作。对于我们模型的有限秩扰动，我们在变形的 Wigner 系综的经典位置找到异常值。