Eugene Strahov

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.

Hard edge limit of the product of two strongly coupled random matrices

We investigate the hard edge scaling limit of the ensemble defined by the squared singular values of the product of two coupled complex random matrices. When taking the coupling parameter to be dependent on the size of the product matrix, in a certain double scaling regime at the origin the two matrices become strongly coupled and we obtain a new hard edge limiting kernel. It interpolates between the classical Bessel-kernel describing the hard edge scaling limit of the Laguerre ensemble of a single matrix, and the Meijer G-kernel of Kuijlaars and Zhang describing the hard edge scaling limit for the product of two independent Gaussian complex matrices. It differs from the interpolating kernel of Borodin to which we compare as well.

Dropping the Independence: Singular Values for Products of Two Coupled Random Matrices

We study the singular values of the product of two coupled rectangular random matrices as a determinantal point process. Each of the two factors is given by a parameter dependent linear combination of two independent, complex Gaussian random matrices, which is equivalent to a coupling of the two factors via an Itzykson-Zuber term. We prove that the squared singular values of such a product form a biorthogonal ensemble and establish its exact solvability. The parameter dependence allows us to interpolate between the singular value statistics of the Laguerre ensemble and that of the product of two independent complex Ginibre ensembles which are both known. We give exact formulae for the correlation kernel in terms of a complex double contour integral, suitable for the subsequent asymptotic analysis. In particular, we derive a Christoffel–Darboux type formula for the correlation kernel, based on a five term recurrence relation for our biorthogonal functions. It enables us to find its scaling limit at the origin representing a hard edge. The resulting limiting kernel coincides with the universal Meijer G-kernel found by several authors in different ensembles. We show that the central limit theorem holds for the linear statistics of the singular values and give the limiting variance explicitly.

Correlation Kernels for Discrete Symplectic and Orthogonal Ensembles

In [49] H. Widom derived formulae expressing correlation functions of orthogonal and symplectic ensembles of random matrices in terms of orthogonal polynomials. We obtain similar results for discrete ensembles with rational discrete logarithmic derivative, and compute explicitly correlation kernels associated to the classical Meixner and Charlier orthogonal polynomials.

Matrix Kernels for Measures on Partitions

We consider the problem of computation of the correlation functions for the z-measures with the deformation (Jack) parameters 2 or 1/2. Such measures on partitions are originated from the representation theory of the infinite symmetric group, and in many ways are similar to the ensembles of Random Matrix Theory of β=4 or β=1 symmetry types. For a certain class of such measures we show that correlation functions can be represented as Pfaffians including 2×2 matrix valued kernels, and compute these kernels explicitly. We also give contour integral representations for correlation kernels of closely connected measures on partitions.

Full length article: A universality theorem for ratios of random characteristic polynomials

We consider asymptotics of ratios of random characteristic polynomials associated with orthogonal polynomial ensembles. Under some natural conditions on the measure in the definition of the orthogonal polynomial ensemble we establish a universality limit for these ratios.

Product Matrix Processes for Coupled Multi-Matrix Models and Their Hard Edge Scaling Limits

Product matrix processes are multi-level point processes formed by the singular values of random matrix products. In this paper, we study such processes where the products of up to m complex random matrices are no longer independent, by introducing a coupling term and potentials for each product. We show that such a process still forms a multi-level determinantal point processes, and give formulae for the relevant correlation functions in terms of the corresponding kernels. For a special choice of potential, leading to a Gaussian coupling between the mth matrix and the product of all previous

The z-measures on partitions, Pfaffian point processes, and the matrix hypergeometric kernel

m-1