Singular values for products of two coupled random matrices: hard edge phase transition

Abstract

Consider the product $GX$ of two rectangular complex random matrices coupled by a constant matrix $\Omega$, where $G$ can be thought to be a Gaussian matrix and $X$ is a bi-invariant polynomial ensemble. We prove that the squared singular values form a biorthogonal ensemble in Borodin's sense, and further that for $X$ being Gaussian the correlation kernel can be expressed as a double contour integral. When all but finitely many eigenvalues of ${\Omega }^{}{\Omega }^{\ast }$ are equal, the corresponding correlation kernel is shown to admit a phase transition phenomenon at the hard edge in four different regimes as the coupling matrix changes. Specifically, the four limiting kernels in turn are the Meijer G-kernel for products of two independent Gaussian matrices, a new critical and interpolating kernel, the perturbed Bessel kernel and the finite coupled product kernel associated with $GX$. In the special case that $X$ is also a Gaussian matrix and $\Omega$ is scalar, such a product has been recently investigated by Akemann and Strahov. We also propose a Jacobi-type product and prove the same transition.

Proof of Theorem 1.3 with more details, 35 pages, Constr Approx 2017

References in corpus (10)

• Multivariate analysis and Jacobi ensembles: largest eigenvalue, Tracy--Widom limits and rates of convergence
• Recent exact and asymptotic results for products of independent random matrices
• Airy kernel with two sets of parameters in directed percolation and random matrix theory
• Singular value statistics of matrix products with truncated unitary matrices
• A new Chiral Two-Matrix Theory for Dirac Spectra with Imaginary Chemical Potential
• Differential equations for singular values of products of Ginibre random matrices
• Singular values for products of complex Ginibre matrices with a source: hard edge limit and phase transition
• Hard edge limit of the product of two strongly coupled random matrices
• Dropping the independence: singular values for products of two coupled random matrices
• Gaussian perturbations of hard edge random matrix ensembles

Cited by in corpus (5)

• Matrix product ensembles of Hermite-type and the hyperbolic Harish-Chandra-Itzykson-Zuber integral
• Finite rank perturbations in products of coupled random matrices: From one correlated to two Wishart ensembles
• Hard Edge Statistics of Products of Pólya Ensembles and Shifted GUE's
• Large n limit for the product of two coupled random matrices
• Product matrix processes for coupled multi-matrix models and their hard edge scaling limits