论文标题
来自Diophantine方程的奇异值的当地法律
A Local Law for Singular Values from Diophantine Equations
论文作者
论文摘要
我们介绍$ n \ times n $随机矩阵$$ x_ {j，k} = \ exp \ left（2πi\ sum_ {q = 1}^d \ω_________________{j，q} k^q \ right） j \ leq n \\ 1 \ leq q \ leq d}} \ text {i.i.d.随机变量}，$$和$ d $ a固定整数。我们证明，其奇异值的分布在显式，小$θ_d> 0 $的情况下以尺度$ n^{ - θ_d} $收敛到本地Marchenko-Pastur定律。据我们所知，这是一个随机矩阵合奏的第一个实例，该集合仅根据$ o（n）$随机变量而明确定义，随机变量显示了通用局部频谱定律。我们的主要技术贡献是在同时考虑到随机和振荡取消的稳定性转换的浓度界限。我们证明的重要成分是对二磷剂方程的解决方案数量的强有力估计（以vinogradov最近的主要猜想的形式出现在Bourgain-Demeter-Guth最近证明）和Pigeonhole的论点，将病房的身份与来自Newton-Girard身份衍生的二磷酸方程的代数独特条件相结合。
We introduce the $N\times N$ random matrices $$ X_{j,k}=\exp\left(2πi \sum_{q=1}^d\ ω_{j,q} k^q\right) \quad \text{with } \{ω_{j,q}\}_{\substack{1\leq j\leq N\\ 1\leq q\leq d}} \text{ i.i.d. random variables}, $$ and $d$ a fixed integer. We prove that the distribution of their singular values converges to the local Marchenko-Pastur law at scales $N^{-θ_d}$ for an explicit, small $θ_d>0$, as long as $d\geq 18$. To our knowledge, this is the first instance of a random matrix ensemble that is explicitly defined in terms of only $O(N)$ random variables exhibiting a universal local spectral law. Our main technical contribution is to derive concentration bounds for the Stieltjes transform that simultaneously take into account stochastic and oscillatory cancellations. Important ingredients in our proof are strong estimates on the number of solutions to Diophantine equations (in the form of Vinogradov's main conjecture recently proved by Bourgain-Demeter-Guth) and a pigeonhole argument that combines the Ward identity with an algebraic uniqueness condition for Diophantine equations derived from the Newton-Girard identities.
