Singular Values of Products of Ginibre Random Matrices
Singular Values of Products of Ginibre Random Matrices
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DOI：
10.1111/sapm.12147
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发表时间：
2016-05
影响因子：
2.7
通讯作者：
N. Witte;P. Forrester
中科院分区：
文献类型：
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作者：
N. Witte;P. Forrester
M个复Ginibre矩阵的乘积的平方奇异值形成双正交系综，因此它们的分布完全由相关核确定。核函数允许硬边缩放到某些Meijer G函数或等价的超几何函数0FM（也称为超贝塞尔函数）的形式。在M=1的情况下，众所周知，对于（0，s）中没有平方奇异值的相应间隙概率可以根据Painlevé III'系统的特定sigma形式的解来计算。这个结果的一种方法是由于Tracy和Widom的形式主义，涉及减少一定的可积系统。Strahov把这个形式推广到一般的M≥1，但没有展示它的约化。在详细介绍了M=1时的必要工作之后，我们考虑了将M=2时的12个耦合微分方程化为一个微分方程的预解式的问题。对于一般硬边参数，发现了一个显式的四阶非线性。对于一个特定的参数选择，证据表明，这简化为一个更简单的三阶非线性方程。讨论了四阶方程的小和大s渐近性，以及M=2系统与所谓的四维Painlevé型方程的可能关系。
The squared singular values of the product of M complex Ginibre matrices form a biorthogonal ensemble, and thus their distribution is fully determined by a correlation kernel. The kernel permits a hard edge scaling to a form specified in terms of certain Meijer G‐functions, or equivalently hypergeometric functions 0FM , also referred to as hyper‐Bessel functions. In the case M=1, it is well known that the corresponding gap probability for no squared singular values in (0, s) can be evaluated in terms of a solution of a particular sigma form of the Painlevé III' system. One approach to this result is a formalism due to Tracy and Widom, involving the reduction of a certain integrable system. Strahov has generalized this formalism to general M≥1 , but has not exhibited its reduction. After detailing the necessary working in the case M=1 , we consider the problem of reducing the 12 coupled differential equations in the case M=2 to a single differential equation for the resolvent. An explicit fourth‐order nonlinear is found for general hard edge parameters. For a particular choice of parameters, evidence is given that this simplifies to a much simpler third‐order nonlinear equation. The small and large s asymptotics of the fourth‐order equation are discussed, as is a possible relationship of the M=2 systems to so‐called four‐dimensional Painlevé‐type equations.