论文标题
来自真实零的随机多项式的随机全部功能
Random entire functions from random polynomials with real zeros
论文作者
论文摘要
我们指出了一个简单的标准，用于将多项式收敛到Laguerre-Pólya（$ \ Mathcal {lp} $）类中的混凝土全部功能（在所有功能中，所有功能都以统一的多项式限制（仅具有真实根）而产生）。然后，我们使用它来证明任何随机$ \ Mathcal {lp} $函数都可以作为无限单位不变的随机遗传矩阵的主要子膜的重新定性多项式的均匀极限。相反，任何无限的随机单位不变的Hermitian矩阵的主要子矩阵的重新缩放特性多项式均匀地收敛到随机$ \ Mathcal {lp} $函数。该结果还具有自然扩展到$β$浓度。杰出的情况包括与$β$ -Sine相关的随机整个功能，更通常是$β$ -HUA-PICKRELL，$β$ -Bessel和$β$ -Airy点过程中研究了文献中研究的过程。
We point out a simple criterion for convergence of polynomials to a concrete entire function in the Laguerre-Pólya ($\mathcal{LP}$) class (of all functions arising as uniform limits of polynomials with only real roots). We then use this to show that any random $\mathcal{LP}$ function can be obtained as the uniform limit of rescaled characteristic polynomials of principal submatrices of an infinite unitarily invariant random Hermitian matrix. Conversely, the rescaled characteristic polynomials of principal submatrices of any infinite random unitarily invariant Hermitian matrix converge uniformly to a random $\mathcal{LP}$ function. This result also has a natural extension to $β$-ensembles. Distinguished cases include random entire functions associated to the $β$-Sine, and more generally $β$-Hua-Pickrell, $β$-Bessel and $β$-Airy point processes studied in the literature.
