Evidence of Random Matrix Corrections for the Large Deviations of Selberg’s Central Limit Theorem
Evidence of Random Matrix Corrections for the Large Deviations of Selberg’s Central Limit Theorem
复制标题
塞尔伯格中心极限定理大偏差的随机矩阵修正的证据
DOI：
10.1080/10586458.2021.2011806
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发表时间：
2021
影响因子：
0.5
通讯作者：
Rao, R.
中科院分区：
文献类型：
--
作者：
Amzallag, E.;Arguin, L.-P.;Bailey, E.;Hui, K.;Rao, R.
Selberg的中心极限定理指出，的值，其中τ是一个均匀随机变量，像均值为0和标准差的高斯随机变量一样渐近分布。根据Radziwiłł的推测，这种分布在阶值下会被打破，在k> 0的水平上会出现一个乘法修正k。这个常数应该与Keating和Snaith对ζ的矩的渐近导所推测的常数相同。在本文中，我们为这一猜想提供了数值和理论证据。我们认为，这种修正对大小区间的最大值的分布有显著的影响。预测的精度使低阶t的数值检测成为可能。一个类似的修正出现在随机酉矩阵特征多项式对数的Keating-Snaith中心极限定理的大偏差中，这是由fsamray， msamliot和Nikeghbali首先证明的。
Selberg’s central limit theorem states that the values of, whereτis a uniform random variable on, are asymptotically distributed like a Gaussian random variable of mean 0 and standard deviation. It was conjectured by Radziwiłł that this distribution breaks down for values of order, where a multiplicative correctionCkwould be present at level,k> 0. This constant should be the same as the one conjectured by Keating and Snaith for the leading asymptotic of themoment ofζ. In this paper, we provide numerical and theoretical evidence for this conjecture. We propose that this correction has a significant effect on the distribution of the maximum ofin intervals of size. The precision of the prediction enables the numerical detection ofCkeven for lowT’s of order. A similar correction appears in the large deviations of the Keating–Snaith central limit theorem for the logarithm of the characteristic polynomial of a random unitary matrix, as first proved by Féray, Méliot and Nikeghbali.