Dropping the independence: singular values for products of two coupled random matrices

Akemann G, Strahov E. Dropping the independence: singular values for products of two coupled random matrices. COMMUNICATIONS IN MATHEMATICAL PHYSICS. 2016;345(1):101-140.We study the singular values of the product of two coupled rectangular randommatrices as a determinantal point process. Each of the two factors is given bya parameter dependent linear combination of two independent, complex Gaussianrandom matrices, which is equivalent to a coupling of the two factors via anItzykson-Zuber term. We prove that the squared singular values of such aproduct form a biorthogonal ensemble and establish its exact solvability. Theparameter dependence allows us to interpolate between the singular valuestatistics of the Laguerre ensemble and that of the product of two independentcomplex Ginibre ensembles which are both known. We give exact formulae for thecorrelation kernel in terms of a complex double contour integral, suitable forthe subsequent asymptotic analysis. In particular, we derive aChristoffel-Darboux type formula for the correlation kernel, based on a fiveterm recurrence relation for our biorthogonal functions. It enables us to findits scaling limit at the origin representing a hard edge. The resultinglimiting kernel coincides with the universal Meijer G-kernel found by severalauthors in different ensembles. We show that the central limit theorem holdsfor the linear statistics of the singular values and give the limiting varianceexplicitly