论文标题
高斯$β$ - 安装特征多项式的强近似：边缘状态和随机通风函数
Strong approximation of Gaussian $β$-ensemble characteristic polynomials: the edge regime and the stochastic Airy function
论文作者
论文摘要
我们通过其转移矩阵复发研究了高斯$β$ for General $β> 0 $的特征多项式。我们表明，重新验证的特征多项式会在极限光谱边缘附近的随机整个功能收敛。这种随机的整个功能（称为随机通风函数）是独特的（缩放）$ l^2 $解决随机通风方程，这是一个二阶随机微分方程的家族。此外，我们获得了特征多项式和随机通风方程的解决方案之间的耦合，这使我们能够证明，对于任何$ε> 0 $，这两个函数在$ n^{ - 1/6 +ε} $上均匀地关闭，并且具有压倒性的可能性。这些结果基于作者的结果，其中分析了特征多项式的转移矩阵复发的双曲线部分。
We investigate the characteristic polynomials of the Gaussian $β$-ensemble for general $β>0$ through its transfer matrix recurrence. We show that the rescaled characteristic polynomial converges to a random entire function in a neighborhood of the edge of the limiting spectrum. This random entire function, called the stochastic Airy function, is the unique (up to scaling) $L^2$ solution to the stochastic Airy equation, a family of second order stochastic differential equations. Moreover, we obtain a coupling between the characteristic polynomial and a solution of the stochastic Airy equation which allows us to show that for any $ε>0$, these two function are uniformly close by $N^{-1/6 + ε}$ with overwhelming probability. These results build on the results of the authors in which the hyperbolic portion of the transfer matrix recurrence for the characteristic polynomial is analyzed.
