Universal objects of the infinite beta random matrix theory
Universal objects of the infinite beta random matrix theory
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DOI：
10.4171/jems/1336
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发表时间：
2020-09
影响因子：
2.6
通讯作者：
V. Gorin;V. Kleptsyn
中科院分区：
文献类型：
--
作者：
V. Gorin;V. Kleptsyn
我们开发了一个理论的多层分布的特征值补充戴森的三倍$\beta= 1，2，4 $的方法对应于真实的/复杂/四元数矩阵的$\beta=\infty$点。我们的中心对象是G$\infty$E系综，它是经典高斯正交/酉/辛系综的对应物，以及Airy$_{\infty}$线系综，它是一组连续曲线，作为$\beta=\infty$处最大特征值的缩放极限。我们对这些物体提出了两种观点。人们可能把它们看作是收集白色噪声的某些加性聚合物的配分函数。可积的观点通过所谓的伴随Hermite多项式和Airy函数的积分来表示它们的分布。我们还概述了我们的合奏作为缩放限制的普遍外观。
We develop a theory of multilevel distributions of eigenvalues which complements the Dyson's threefold $\beta=1,2,4$ approach corresponding to real/complex/quaternion matrices by $\beta=\infty$ 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 $\beta=\infty$. 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.