Product of Ginibre matrices: Fuss-Catalan and Raney distributions.
Product of Ginibre matrices: Fuss-Catalan and Raney distributions.
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DOI：
10.1103/physreve.83.061118
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发表时间：
2011-03
期刊：
影响因子：
--
通讯作者：
K. Penson;K. Życzkowski
中科院分区：
文献类型：
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作者：
K. Penson;K. Życzkowski
s阶随机Ginibre矩阵乘积的平方奇异值渐近地由概率分布P（s）（x）刻画，使得它们的矩等于s阶Fuss-Catalan数。我们找到了Fuss-Catalan分布P（s）（x）的一个表示，它表示为（s）F（s-1）型的s个超几何函数的组合.这里导出的显式公式对于任意正整数s是精确的，并且对于s=1，它归结为Marchenko-Pastur分布。使用类似的技术，涉及梅林变换和梅杰G函数，我们找到确切的表达的阮内概率分布，这是由两个参数的Fuss-Catalan数的推广的时刻。这些分布也可以被认为是维格纳循环定律的双参数推广。
Squared singular values of a product of s square random Ginibre matrices are asymptotically characterized by probability distributions P(s)(x), such that their moments are equal to the Fuss-Catalan numbers of order s. We find a representation of the Fuss-Catalan distributions P(s)(x) in terms of a combination of s hypergeometric functions of the type (s)F(s-1). The explicit formula derived here is exact for an arbitrary positive integer s, and for s=1 it reduces to the Marchenko-Pastur distribution. Using similar techniques, involving the Mellin transform and the Meijer G function, we find exact expressions for the Raney probability distributions, the moments of which are given by a two-parameter generalization of the Fuss-Catalan numbers. These distributions can also be considered as a two-parameter generalization of the Wigner semicircle law.