评注：关于“对称伪随机矩阵”
Comment on "Symmetric Pseudo-Random Matrices"
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中文总结 AI 辅助
本文指出Soloveychik等人关于对称循环矩阵谱分布证明中的技术缺陷，并利用Katz定理证明其特征值为扭曲Kloosterman和，从而确定性建立半圆律。
中文摘要 AI 辅助
2018年，Soloveychik、Xiang和Tarokh考虑了一个由长度为$n=2^m-1$的二元Golomb序列构造的伪随机对称循环矩阵，并声称通过矩方法证明了其经验谱分布随着$n$趋于无穷几乎必然收敛于半圆律。在本评注中，我们指出他们的论证包含若干技术缺陷以及对技术引理的不当应用。相反，我们证明该矩阵的特征值本质上是一个随乘法特征变化的归一化扭曲Kloosterman和，对此我们直接应用Katz的结果，以确定性地建立渐近半圆谱分布。
英文摘要
In 2018, Soloveychik, Xiang and Tarokh considered a pseudo-random symmetric circulant matrix constructed from binary Golomb sequences of length $n=2^m-1$, and claimed a proof that its empirical spectral distribution converges almost surely to the semicircle law as $n$ grows to infinity by the method of moments. In this comment note we show that their argument contains several technical flaws and inappropriate applications of technical lemmas. Instead we demonstrate that the eigenvalues of the matrix are simply a normalized twisted Kloosterman sum varying over the multiplicative character, to which we apply Katz's result directly to establish the asymptotically semicircle spectral distribution deterministically.
发表机构
- Hetao Institute of Mathematics and Interdisciplinary Sciences (HIMIS)（河套数学与交叉科学研究院）
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