Permanental processes from products of complex and quaternionic induced Ginibre ensembles

We consider products of independent random matrices taken from the induced Ginibre ensemble with complex or quaternion elements. The joint densities for the complex eigenvalues of the product matrix can be written down exactly for a product of any fixed number of matrices and any finite matrix size. We show that the squared absolute values of the eigenvalues form a permanental process, generalising the results of Kostlan and Rider for single matrices to products of complex and quaternionic matrices. Based on these findings, we can first write down exact results and asymptotic expansions for the so-called hole probabilities, that a disc centered at the origin is void of eigenvalues. Second, we compute the asymptotic expansion for the opposite problem, that a large fraction of complex eigenvalues occupies a disc of fixed radius centered at the origin; this is known as the overcrowding problem. While the expressions for finite matrix size depend on the parameters of the induced ensembles, the asymptotic results agree to leading order with previous results for products of square Ginibre matrices.Comment: 47 pages, v2: typos corrected, 1 reference added, published versio

An exact formula for general spectral correlation function of random Hermitian matrices

We have found an exact formula expressing a general correlation function containing both products and ratios of characteristic polynomials of random Hermitian matrices. The answer is given in the form of a determinant. An essential difference from the previously studied correlation functions (of products only) is the appearance of non-polynomial functions along with the orthogonal polynomials. These non-polynomial functions are the Cauchy transforms of the orthogonal polynomials. The result is valid for any ensemble of beta=2 symmetry class and generalizes recent asymptotic formulae obtained for GUE and its chiral counterpart by different methods..Comment: published version, with a few misprints correcte

Characteristic Polynomials of Sample Covariance Matrices: The Non-Square Case

We consider the sample covariance matrices of large data matrices which have i.i.d. complex matrix entries and which are non-square in the sense that the difference between the number of rows and the number of columns tends to infinity. We show that the second-order correlation function of the characteristic polynomial of the sample covariance matrix is asymptotically given by the sine kernel in the bulk of the spectrum and by the Airy kernel at the edge of the spectrum. Similar results are given for real sample covariance matrices

Random Matrices close to Hermitian or unitary: overview of methods and results

The paper discusses progress in understanding statistical properties of complex eigenvalues (and corresponding eigenvectors) of weakly non-unitary and non-Hermitian random matrices. Ensembles of this type emerge in various physical contexts, most importantly in random matrix description of quantum chaotic scattering as well as in the context of QCD-inspired random matrix models.Comment: Published version, with a few more misprints correcte

Products of Independent Quaternion Ginibre Matrices and their Correlation Functions

Ipsen J. Products of Independent Quaternion Ginibre Matrices and their Correlation Functions. J. Phys. A: Math. Theor. 2013;46(26): 265201.We discuss the product of independent induced quaternion ($\beta=4$) Ginibrematrices, and the eigenvalue correlations of this product matrix. The jointprobability density function for the eigenvalues of the product matrix is shownto be identical to that of a single Ginibre matrix, but with a more complicatedweight function. We find the skew-orthogonal polynomials corresponding to theweight function of the product matrix, and use the method of skew-orthogonalpolynomials to compute the eigenvalue correlation functions for productmatrices of finite size. The radial behavior of the density of states isstudied in the limit of large matrices, and the macroscopic density isdiscussed. The microscopic limit at the origin, at the edge(s) and in the bulkis also discussed for the radial behavior of the density of states

Hole probabilities and overcrowding estimates for products of complex Gaussian matrices

Akemann G, Strahov E. Hole probabilities and overcrowding estimates for products of complex Gaussian matrices. Journal of Statistical Physics. 2013;151(6):987-1003.We consider eigenvalues of a product of n non-Hermitian, independent randommatrices. Each matrix in this product is of size N\times N with independentstandard complex Gaussian variables. The eigenvalues of such a product form adeterminantal point process on the complex plane (Akemann and Burda J. Phys A:Math. Theor. 45 (2012) 465201), which can be understood as a generalization ofthe finite Ginibre ensemble. As N\rightarrow\infty, a generalized infiniteGinibre ensemble arises. We show that the set of absolute values of the pointsof this determinantal process has the same distribution as{R_1^{(n)},R_2^{(n)},...}, where R_k^{(n)} are independent, and (R_k^{(n)})^2is distributed as the product of n independent Gamma variables Gamma(k,1). Thisenables us to find the asymptotics for the hole probabilities, i.e. for theprobabilities of the events that there are no points of the process in a discof radius r with its center at 0, as r\rightarrow\infty. In addition, we solvethe relevant overcrowding problem: we derive an asymptotic formula for theprobability that there are more than m points of the process in a fixed disk ofradius r with its center at 0, as m\rightarrow\infty