论文标题
无限β随机矩阵理论的通用对象
Universal objects of the infinite beta random matrix theory
论文作者
论文摘要
我们开发了特征值的多级分布理论，该理论补充了戴森的三倍$β= 1,2,4 $方法，与$β= \ infty $ point相对于真实/复杂/Quaternion矩阵。我们的中心对象是g $ \ infty $ e Ensemble，它是古典高斯正交/统一/符合/符号合奏的对应物，以及通风$ _ {\ infty} $ line Ensemble，这是连续曲线的集合，可作为$β= \ ifty $ biptle eigenvalues的缩放率限制。我们对这些对象产生了两个观点。概率一个人将它们视为某些收集白噪声的某些加性聚合物的分区功能。可集成的观点通过所谓的相关的Hermite多项式和通风功能的积分表达了它们的分布。我们还概述了合奏的普遍外观为缩放限制。
We develop a theory of multilevel distributions of eigenvalues which complements the Dyson's threefold $β=1,2,4$ approach corresponding to real/complex/quaternion matrices by $β=\infty$ point. Our central objects are G$\infty$E ensemble, which is a counterpart of classical Gaussian Orthogonal/Unitary/Symplectic ensembles, and Airy$_{\infty}$ line ensemble, which is a collection of continuous curves serving as a scaling limit for largest eigenvalues at $β=\infty$. We develop two points of views on these objects. Probabilistic one treats them as partition functions of certain additive polymers collecting white noise. Integrable point of view expresses their distributions through the so-called associated Hermite polynomials and integrals of Airy function. We also outline universal appearances of our ensembles as scaling limits.
