丢弃独立性：两个耦合随机矩阵乘积的奇异值
Dropping the independence: singular values for products of two coupled random matrices
摘要：
研究了两个耦合矩形随机矩阵乘积作为一个行列式点过程的奇异值问题。这两个因素中的每一个都由两个独立的复高斯随机矩阵的参数相关线性组合给出，这相当于通过Itzykson-Zuber项耦合这两个因素。我们证明了这样的产品的平方奇异值形成一个双正交系综，并建立其精确的可解性。的参数依赖性允许我们插值之间的奇异值统计的Laguerre合奏和两个独立的复杂的Ginibre合奏的产品都是已知的。我们给出了精确的公式的相关核在一个复杂的双围道积分，适用于随后的渐近分析。特别是，我们推导出的相关核的Christoffel-Darboux型公式的基础上，我们的双正交函数的五项递归关系。它使我们能够在代表硬边的原点处找到其缩放极限。由此产生的限制内核与几位作者在不同的合奏中发现的通用Meijer G-内核相吻合。我们证明了奇异值线性统计量的中心极限定理成立，并明确给出了极限方差。
英文摘要：
We study the singular values of the product of two coupled rectangular random matrices as a determinantal point process. Each of the two factors is given by a parameter dependent linear combination of two independent, complex Gaussian random matrices, which is equivalent to a coupling of the two factors via an Itzykson-Zuber term. We prove that the squared singular values of such a product form a biorthogonal ensemble and establish its exact solvability. The parameter dependence allows us to interpolate between the singular value statistics of the Laguerre ensemble and that of the product of two independent complex Ginibre ensembles which are both known. We give exact formulae for the correlation kernel in terms of a complex double contour integral, suitable for the subsequent asymptotic analysis. In particular, we derive a Christoffel-Darboux type formula for the correlation kernel, based on a five term recurrence relation for our biorthogonal functions. It enables us to find its scaling limit at the origin representing a hard edge. The resulting limiting kernel coincides with the universal Meijer G-kernel found by several authors in different ensembles. We show that the central limit theorem holds for the linear statistics of the singular values and give the limiting variance explicitly.
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