Dropping the Independence: Singular Values for Products of Two Coupled Random Matrices

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• 作者：Gernot Akemann ; Eugene Strahov
• 刊名：Communications in Mathematical Physics
• 出版年：2016
• 出版时间：July 2016
• 年：2016
• 卷：345
• 期：1
• 页码：101-140
• 全文大小：735 KB
• 刊物类别：Physics and Astronomy
• 刊物主题：Physics
  Mathematical and Computational Physics
  Quantum Physics
  Quantum Computing, Information and Physics
  Complexity
  Statistical Physics
  Relativity and Cosmology
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-0916
• 卷排序：345

文摘

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.G. Akemann is supported partly by Investissements d’Avenir du LabEx PALM (ANR-10-LABX-0039-PALM) and by the SFB|TR12 “Symmetries and Universality in Mesoscopic Systems” of the German research council DFG. E. Strahov is supported in part by the Hebrew University Grant “Non-hermitian random matrices” No. 0337592.Communicated by P. Deift