Complex Ginibre Matrices Research Articles

Characteristic polynomials of products of non-Hermitian Wigner matrices: finite-N results and Lyapunov universality

We compute the average characteristic polynomial of the hermitised product of $M$ real or complex Wigner matrices of size $N\times N$ and the average of the characteristic polynomial of a product of $M$ such Wigner matrices times the characteristic polynomial of the conjugate matrix. Surprisingly, the results agree with that of the product of $M$ real or complex Ginibre matrices at finite-$N$, which have i.i.d. Gaussian entries. For the latter the average characteristic polynomial yields the orthogonal polynomial for the singular values of the product matrix, whereas the product of the two characteristic polynomials involves the kernel of complex eigenvalues. This extends the result of Forrester and Gamburd for one characteristic polynomial of a single random matrix and only depends on the first two moments. In the limit $M\to\infty$ at fixed $N$ we determine the locations of the zeros of a single characteristic polynomial, rescaled as Lyapunov exponents by taking the logarithm of the $M$th root. The position of the $j$th zero agrees asymptotically for large-$j$ with the position of the $j$th Lyapunov exponent for products of Gaussian random matrices, hinting at the universality of the latter.

Schwinger–Dyson and loop equations for a product of square Ginibre random matrices

In this paper, we study the product of two complex Ginibre matrices and the loop equations satisfied by their resolvents (i.e. the Stieltjes transform of the correlation functions). We obtain using Schwinger–Dyson equation (SDE) techniques the general loop equations satisfied by the resolvents. In order to deal with the product structure of the random matrix of interest, we consider SDEs involving the integral of higher derivatives. One of the advantage of this technique is that it bypasses the reformulation of the problem in terms of singular values. As a byproduct of this study we obtain the large N limit of the Stieltjes transform of the 2-point correlation function, as well as the first correction to the Stieltjes transform of the density, giving us access to corrections to the smoothed density. In order to pave the way for the establishment of a topological recursion formula we also study the geometry of the corresponding spectral curve. This paper also contains explicit results for different resolvents and their corrections.

Eigenvector statistics of the product of Ginibre matrices.

We develop a method to calculate left-right eigenvector correlations of the product of m independent N×N complex Ginibre matrices. For illustration, we present explicit analytical results for the vector overlap for a couple of examples for small m and N. We conjecture that the integrated overlap between left and right eigenvectors is given by the formula O=1+(m/2)(N-1) and support this conjecture by analytical and numerical calculations. We derive an analytical expression for the limiting correlation density as N→∞ for the product of Ginibre matrices as well as for the product of elliptic matrices. In the latter case, we find that the correlation function is independent of the eccentricities of the elliptic laws.

Universality for Products of Random Matrices I: Ginibre and Truncated Unitary Cases

Recently, the joint probability density functions of complex eigenvalues for products of independent complex Ginibre matrices have been explicitly derived as determinantal point processes. We express truncated series coming from the correlation kernels as multivariate integrals with singularity and investigate saddle point method for such a type of integrals. As an application, we prove that the eigenvalue correlation functions have the same scaling limits as those of the single complex Ginibre ensemble, both in the bulk and at the edge of the spectrum. We also prove that the similar results hold true for products of independent truncated unitary matrices.

Singular values of products of random matrices and polynomial ensembles

Akemann, Ipsen, and Kieburg showed recently that the squared singular values of a product of M complex Ginibre matrices are distributed according to a determinantal point process. We introduce the notion of a polynomial ensemble and show how their result can be interpreted as a transformation of polynomial ensembles. We also show that the squared singular values of the product of M - 1 complex Ginibre matrices with one truncated unitary matrix is a polynomial ensemble, and we derive a double integral representation for the correlation kernel associated with this ensemble. We use this to calculate the scaling limit at the hard edge, which turns out to be the same scaling limit as the one found by Kuijlaars and Zhang for the squared singular values of a product of M complex Ginibre matrices. Our final result is that these limiting kernels also appear as scaling limits for the biorthogonal ensembles of Borodin with parameter θ > 0, in case θ or 1/θ is an integer. This further supports the conjecture that these kernels have a universal character.