Singular Values for Products of Two Coupled Random Matrices: Hard Edge Phase Transition

Abstract

Abstract. Consider the product GX of two rectangular complex random matrices coupled by a constant matrix Ω, 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 ΩΩ * 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 Ω 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.

Additive matrix convolutions of Pólya ensembles and polynomial ensembles

“…The first case is for a general Pólya ensemble without a shift on either the Hermitian antisymmetric matrices H 1 , the Hermitian matrices H 2 , the Hermitian anti-self-dual matrices H 4 or the complex rectangular matrices M ν . The other two cases we considered are the eigenvalue/squared singular value statistics of the Pólya ensemble added by a either a fixed matrix or a random matrix drawn from a polynomial ensemble on the same space as the Pólya ensemble, see [4,14,38,[41][42][43][44]. All results hold for finite matrix dimension.…”

Finite rank perturbations in products of coupled random matrices: From one correlated to two Wishart ensembles

“…A further direction was taken in [21] by adding an external field to the product of r Gaussian matrices. All these deformations allow to study finite rank perturbations of the known Bessel and Meijer Gkernel, extending the results of [15] for a single Wishart matrix with external field for the former, and of [21,35] for the latter. Similar findings were made earlier for deformations of the Airy kernel at the soft edge [8,15,13], where a relation to directed percolation was pointed out.…”

“…truncated unitary matrices [29], the question is open for products of matrices from unitary bi-invariant ensembles with general distributions. The difficulty is that the unitary group integrals needed after singular value decomposition are not available in general (see however [35]).…”

“…This is the route we will choose here, for the product of r = 2 matrices as a starting point. We will combine this with a coupling among the two random matrices of scalar [5,6] or matrix-valued type [35], investigating the most general distribution that is quadratic in the two random matrices and remains determinantal. A further direction was taken in [21] by adding an external field to the product of r Gaussian matrices.…”

“…In [6] three different scaling limits were identified at the origin of the spectrum representing a hard edge, with the following limiting kernels: (I) the Meijer G-kernel for two independent matrices, proving its universality for a one-parameter family, (II) a parameter dependent kernel that interpolates between the limit (I) and the limit (III), where the well-known universal Bessel kernel (III) was obtained. In a subsequent paper [35] a full coupling matrix Ω was introduced as in (1.2), while keeping the conditions on W = 1+µ 2µ 1 1 N and Q = 1+µ 2µ 1 1 M as in [5]. There, the kernels in the limits (I), (II) and (III) were extended and finite rank perturbations in the limits (II) and (III) of [6] were found.…”