Random Block Matrices Research Articles

Sparse block-structured random matrices: universality

We study ensembles of sparse block-structured random matrices generated from the adjacency matrix of a Erdös–Renyi random graph with N vertices of average degree Z, inserting a real symmetric d × d random block at each non-vanishing entry. We consider several ensembles of random block matrices with rank r < d and with maximal rank, r = d. The spectral moments of the sparse block-structured random matrix are evaluated for , d finite or infinite, and several probability distributions for the blocks (e.g. fixed trace, bounded trace and Gaussian). Because of the concentration of the probability measure in the limit, the spectral moments are independent of the probability measure of the blocks (with mild assumptions of isotropy, smoothness and sub-Gaussian tails). The effective medium approximation is the limiting spectral density of the sparse block-structured random ensembles with finite rank. Analogous classes of universality hold for the Laplacian sparse block-structured ensemble. The same limiting distributions are obtained using random regular graphs instead of Erdös–Renyi graphs.

Sparse random block matrices

The spectral moments of ensembles of sparse random block matrices are analytically evaluated in the limit of large order. The structure of the sparse matrix corresponds to the Erdös–Renyi random graph. The blocks are i.i.d. random matrices of the classical ensembles GOE or GUE. The moments are evaluated for finite or infinite dimension of the blocks. The correspondences between sets of closed walks on trees and classes of irreducible partitions studied in free probability together with functional relations are powerful tools for analytic evaluation of the limiting moments. They are helpful to identify probability laws for the blocks and limits of the parameters which allow the evaluation of all the spectral moments and of the spectral density.

V-density for eigenvalues of random block matrices with independent blocks whose entries have different variances and expectations

Abstract V-density under Lindeberg condition for the independent blocks of random matrices having different variances and expectations is found.

𝑉-law for random block matrices under the generalized Lindeberg condition

Abstract The V-law under generalized Lindeberg condition for the independent blocks of random matrices having double stochastic matrix of covariances and different expectations of their array is proven.

Largest eigenvalue of large random block matrices: A combinatorial approach

We study the largest eigenvalue of certain block matrices where the number of blocks and the block size both increase with suitable conditions on their relative growth. In one of them, we employ a symmetric block structure with large independent Wigner blocks and in the other we have the Wigner block structure with large independent symmetric blocks. The entries are assumed to be independent and identically distributed with mean [Formula: see text] variance [Formula: see text] with an appropriate growth condition on the moments. Under our conditions the limit spectral distribution of these matrices is the standard semi-circle law. It is natural to ask if the extreme eigenvalues converge to the extreme points of its support, namely [Formula: see text]. We exhibit models where this indeed happens as well as models where the spectral norm converges to [Formula: see text]. Our proofs are based on combinatorial analysis of the behavior of the trace of large powers of the matrix.

On the concentration of random multilinear forms and the universality of random block matrices

The circular law asserts that if $${\mathbf {X}}_n$$ is a $$n \times n$$ matrix with iid complex entries of mean zero and unit variance, then the empirical spectral distribution of $$\frac{1}{\sqrt{n}} {\mathbf {X}}_n$$ converges almost surely to the uniform distribution on the unit disk as $$n$$ tends to infinity. Answering a question of Tao, we prove the circular law for a general class of random block matrices with dependent entries. The proof relies on an inverse-type result for the concentration of linear operators and multilinear forms.

Random block matrices generalizing the classical Jacobi and Laguerre ensembles

In this paper we consider random block matrices which generalize the classical Laguerre ensemble and the Jacobi ensemble. We show that the random eigenvalues of the matrices can be uniformly approximated by the zeros of matrix orthogonal polynomials and obtain a rate for the maximum difference between the eigenvalues and the zeros. This relation between the random block matrices and matrix orthogonal polynomials allows a derivation of the asymptotic spectral distribution of the matrices.

Random Block Matrices and Matrix Orthogonal Polynomials

In this paper we consider random block matrices, which generalize the general beta ensembles recently investigated by Dumitriu and Edelmann (J. Math. Phys. 43:5830–5847, 2002; Ann. Inst. Poincaré Probab. Stat. 41:1083–1099, 2005). We demonstrate that the eigenvalues of these random matrices can be uniformly approximated by roots of matrix orthogonal polynomials which were investigated independently from the random matrix literature. As a consequence, we derive the asymptotic spectral distribution of these matrices. The limit distribution has a density which can be represented as the trace of an integral of densities of matrix measures corresponding to the Chebyshev matrix polynomials of the first kind. Our results establish a new relation between the theory of random block matrices and the field of matrix orthogonal polynomials, which have not been explored so far in the literature.

The Limiting Spectra of Girko’s Block-Matrix

To analyze the limiting spectral distribution of some random block-matrices, Girko (Random Oper. Stoch. Equ. 8(2), 189–194, 2000) uses a system of canonical equations from (An Introduction to Statistical Analysis of Random Arrays. VSP, Utrecht, 1998). In this paper, we use the method of moments to give an integral form for the almost sure limiting spectral distribution of such matrices.

Density of Eigenvalues of Random Normal Matrices with an Arbitrary Potential, and of Generalized Normal Matrices

Following the works by Wiegmann-Zabrodin, Elbau-Felder, Hedenmalm-Maka- rov, and others, we consider the normal matrix model with an arbitrary potential function, and explain how the problem of finding the support domain for the asymptotic eigenvalue density of such matrices (when the size of the matrices goes to infinity) is related to the problem of Hele-Shaw flows on curved surfaces, considered by Entov and the first author in 1990-s. In the case when the potential function is the sum of a rotationally invariant function and the real part of a polynomial of the complex coordinate, we use this relation and the conformal mapping method developed by Entov and the first author to find the shape of the support domain explicitly (up to finitely many undetermined parameters, which are to be found from a finite system of equations). In the case when the rotationally invariant function is |z| 2 , this is done by Wiegmann-Zabrodin and Elbau-Felder. We apply our results to the generalized normal matrix model, which deals with random block matrices that give rise to -representations of the deformed preprojective algebra of the affine quiver of type ˆ Am 1. We show that this model is equivalent to the usual normal matrix model in the large N limit. Thus the conformal mapping method can be applied to find explicitly the support domain for the generalized normal matrix model.

Distribution of the eigenvalues of random block-matrices

The asymptotic behaviour of the eigenvalues of random block-matrices is investigated with block sizes tending to infinity in the same order. In the proofs some extended version of Wigner’s semi-circle law as well as perturbation results for symmetric matrices are used. The paper also deals with the asymptotic distribution of the protruding eigenvalues and some applications to random graphs.