Differential equations for singular values of products of Ginibre random matrices
Differential equations for singular values of products of Ginibre random matrices
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DOI：
10.1088/1751-8113/47/32/325203
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发表时间：
2014-03
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通讯作者：
E. Strahov
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文献类型：
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作者：
E. Strahov
Akemann et al .（2013）证明了这一点。Rev. E 88 052118)， M个复Ginibre随机矩阵乘积的奇异值的平方形成一个行列式点过程，其相关核可用Meijer的g函数表示。Kuijlaars和Zhang （arXiv:1308.1003）最近表明，在光谱边缘，该相关核具有显著的缩放极限K M (x, y) ？>可以理解为随机矩阵理论经典贝塞尔核的推广。本文研究核为K M (x, y) χ J (y) ?的算子的Fredholm行列式。>，其中J是区间的不相交并，J =∪J (a 2j−1,a 2j) ？>, χ J ？>是集合J的特征函数。这个Fredholm行列式等于J不包含由K M （x, y）定义的极限行列式点过程的粒子的概率。>（间隙概率）。我们导出了与Fredholm行列式相关的哈密顿微分，并将它们与Jimbo、Miwa、Môri、Ueno和Sato理论的保一变形方程联系起来。在特殊情况下J = (0, s) ？我们给出了一个用非线性常微分方程组的解表示间隙概率的公式。
It was proved by Akemann et al (2013 Phys. Rev. E 88 052118) that squared singular values of products of M complex Ginibre random matrices form a determinantal point process whose correlation kernel is expressible in terms of Meijerʼs G-functions. Kuijlaars and Zhang (arXiv:1308.1003) recently showed that at the edge of the spectrum, this correlation kernel has a remarkable scaling limit K M ( x , y ) ?> which can be understood as a generalization of the classical Bessel kernel of random matrix theory. In this paper we investigate the Fredholm determinant of the operator with the kernel K M ( x , y ) χ J ( y ) ?> , where J is a disjoint union of intervals, J = ∪ j ( a 2 j − 1 , a 2 j ) ?> , and χ J ?> is the characteristic function of the set J. This Fredholm determinant is equal to the probability that J contains no particles of the limiting determinantal point process defined by K M ( x , y ) ?> (the gap probability). We derive the Hamiltonian differential associated with the corresponding Fredholm determinant, and relate them with the monodromy preserving deformation equations of the Jimbo, Miwa, Môri, Ueno and Sato theory. In the special case J = ( 0 , s ) ?> we give a formula for the gap probability in terms of a solution of a system of nonlinear ordinary differential equations.