We explore the boundaries of sine kernel universality for the eigenvalues of Gaussian perturbations of large deterministic Hermitian matrices. Equivalently, we study for deterministic initial data the time after which Dyson’s Brownian motion exhibits sine kernel correlations. We explicitly describe this time span in terms of the limiting density and rigidity of the initial points. Our main focus lies on cases where the initial density vanishes at an interior point of the support. We show that the time to reach universality becomes larger if the density vanishes faster or if the initial points show less rigidity.

中文翻译：

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Hermitian矩阵的高斯扰动的正弦核普遍性边界

我们探索了大型确定性 Hermitian 矩阵的高斯扰动特征值的正弦核普遍性的边界。等效地，我们研究确定性初始数据，在该时间之后戴森的布朗运动表现出正弦核相关性。我们根据初始点的极限密度和刚性明确地描述了这个时间跨度。我们的主要关注点在于初始密度在支撑的内部点消失的情况。我们表明，如果密度消失得更快或初始点的刚性较小，则达到普遍性的时间会变长。