Optimal Local Law and Central Limit Theorem for $$\beta $$-Ensembles
Optimal Local Law and Central Limit Theorem for $$\beta $$-Ensembles
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DOI：
10.1007/s00220-022-04311-2
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发表时间：
2021-03
影响因子：
2.4
通讯作者：
P. Bourgade;Krishnan Mody;Michel Pain
中科院分区：
文献类型：
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作者：
P. Bourgade;Krishnan Mody;Michel Pain
在设置的一般合奏，我们使用的回路方程的层次证明了本地法律的最佳误差为常数，有效的任何规模，包括微观。当地法律有以下后果。(i)有序粒子的最佳刚性尺度在光谱的大部分是有序的。(ii)粒子的涨落满足一个中心极限定理，协方差对应于一个几何相关的场;特别是每个粒子在大规模上波动。(iii)电势的对数也满足数学相关的中心极限定理。与随机矩阵普遍性方面的许多进展相反，这些结果并不能通过比较来进行。的确，他们是新的高斯合奏。通过比较技术，（ii）和（iii）也适用于维格纳矩阵。
In the setting of generic-ensembles, we use the loop equation hierarchy to prove a local law with optimal error up to a constant, valid on any scale including microscopic. This local law has the following consequences. (i) The optimal rigidity scale of the ordered particles is of orderin the bulk of the spectrum. (ii) Fluctuations of the particles satisfy a central limit theorem with covariance corresponding to a logarithmically correlated field; in particular each particle in the bulk fluctuates on scale. (iii) The logarithm of the electric potential also satisfies a logarithmically correlated central limit theorem. Contrary to much progress on random matrix universality, these results do not proceed by comparison. Indeed, they are new for the Gaussian-ensembles. By comparison techniques, (ii) and (iii) also hold for Wigner matrices.