LEVEL-SPACING DISTRIBUTIONS AND THE AIRY KERNEL
LEVEL-SPACING DISTRIBUTIONS AND THE AIRY KERNEL
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DOI：
10.1007/bf02100489
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发表时间：
1994-01-01
影响因子：
2.4
通讯作者：
WIDOM, H
中科院分区：
文献类型：
--
作者：
TRACY, CA;WIDOM, H
在N × N厄米特矩阵的随机矩阵模型中，缩放“大量频谱”中的水平间距分布函数，然后到达极限N ->无穷大，导致正弦核sin pi（x-y）/pi（x-y）的Fredholm行列式。类似地，在“谱的边缘”处的缩放极限导致艾里核[Ai（x）Ai（y）-Ai '（x）Ai（y）]/（x-y）。在本文中，我们推导出类似的艾里核的下列性质的正弦核：完全可积系统的P.D.E.的发现金宝，三和，森，佐藤;的表达，在一个单一的间隔的情况下，Fredholm行列式方面的Painleve超越;存在的一个交换微分算子;以及事实上，这一运营商可以用于推导渐近，一般n，概率的间隔包含正好n个特征值。
Scaling level-spacing distribution functions in the ''bulk of the spectrum'' in random matrix models of N x N hermitian matrices and then going to the limit N --> infinity leads to the Fredholm determinant of the sine kernel sin pi(x - y)/pi(x - y). Similarly a scaling limit at the ''edge of the spectrum'' leads to the Airy kernel [Ai(x) Ai(y) - Ai'(x) Ai(y)]/(x - y). In this paper we derive analogues for this Airy kernel of the following properties of the sine kernel: the completely integrable system of P.D.E.'s found by Jimbo, Miwa, Mori, and Sato; the expression, in the case of a single interval, of the Fredholm determinant in terms of a Painleve transcendent; the existence of a commuting differential operator; and the fact that this operator can be used in the derivation of asymptotics, for general n, of the probability that an interval contains precisely n eigenvalues.