论文标题
高斯β-元素元素特性多项式的强近似：双曲线状态
Strong approximation of Gaussian beta-ensemble characteristic polynomials: the hyperbolic regime
论文作者
论文摘要
我们通过其传输矩阵复发调查了高斯$β$ for General $β> 0 $的特征多项式$φ_n$。我们的动机是在高斯对数 - 与之相关的字段方面获得$φ_n$的（概率）近似值，以最终推断出其一些优质的渐近特性。我们区分不同类型的转移矩阵，并完全分析复发的双曲线状态。结果，我们获得了$φ_n（z）$和高斯分析函数之间的新耦合，其错误是$ z \ in \ mathbb {c} $均匀的，与半圆法的支持分开。我们将其用作输入，以给出Arxiv：2009.05003边缘的特征多项式的几乎确定缩放限制。还需要在大部分半圆定律内获得类似的强近似值。我们的分析依赖于转移矩阵乘积的中等偏差估计值，在不同情况下，这种方法也可能很有用。
We investigate the characteristic polynomials $φ_N$ of the Gaussian $β$-ensemble for general $β>0$ through its transfer matrix recurrence. Our motivation is to obtain a (probabilistic) approximation for $φ_N$ in terms of a Gaussian log--correlated field in order to ultimately deduce some of its fine asymptotic properties. We distinguish between different types of transfer matrices and analyze completely the hyperbolic regime of the recurrence. As a result, we obtain a new coupling between $φ_N(z)$ and a Gaussian analytic function with an error which is uniform for $z \in \mathbb{C}$ separated from the support of the semicircle law. We use this as input to give the almost sure scaling limit of the characteristic polynomial at the edge in arXiv:2009.05003. This is also required to obtain analogous strong approximations inside of the bulk of the semicircle law. Our analysis relies on moderate deviation estimates for the product of transfer matrices and this approach might also be useful in different contexts.
