Limiting spectral distribution of large dimensional random matrices of linear processes

The limiting spectral distribution (LSD) of large sample radom matrices is derived under dependence conditions. We consider the matrices \(X_{N}T_{N}X_{N}^{\prime}\) , where \(X_{N}\) is a matrix (\(N \times n(N)\)) where the column vectors are modeled as linear processes, and \(T_{N}\) is a real symmetric matrix whose LSD exists. Under some conditions we show that, the LSD of \(X_{N}T_{N}X_{N}^{\prime}\) exists almost surely, as \(N \rightarrow \infty\) and \(n(N)/N \rightarrow c > 0\). Numerical simulations are also provided with the intention to study the convergence of the empirical density estimator of the spectral density of \(X_{N}T_{N}X_{N}^{\prime}\).

Spectral distributions of periodic random matrix ensembles

Koloğlu, Kopp and Miller compute the limiting spectral distribution of a certain class of real random matrix ensembles, known as [Formula: see text]-block circulant ensembles, and discover that it is exactly equal to the eigenvalue distribution of an [Formula: see text] Gaussian unitary ensemble. We give a simpler proof that under very general conditions which subsume the cases studied by Koloğlu–Kopp–Miller, real-symmetric ensembles with periodic diagonals always have limiting spectral distribution equal to the eigenvalue distribution of a finite Hermitian ensemble with Gaussian entries which is a ‘complex version’ of a [Formula: see text] submatrix of the ensemble. We also prove an essentially algebraic relation between certain periodic finite Hermitian ensembles with Gaussian entries, and the previous result may be seen as an asymptotic version of this for real-symmetric ensembles. The proofs show that this general correspondence between periodic random matrix ensembles and finite complex Hermitian ensembles is elementary and combinatorial in nature.

Limiting spectral distribution of the product of truncated Haar unitary matrices

Let [Formula: see text], [Formula: see text], be [Formula: see text] probabilistically independent matrices of order [Formula: see text] (with [Formula: see text]) which are the left-uppermost blocks of [Formula: see text] Haar unitary matrices. Suppose that [Formula: see text] as [Formula: see text], with [Formula: see text]. Using free probability and Brown measure techniques, we find the limiting spectral distribution of [Formula: see text].

Brown measure and asymptotic freeness of elliptic and related matrices

We show that independent elliptic matrices converge to freely independent elliptic elements. Moreover, the elliptic matrices are asymptotically free with deterministic matrices under appropriate conditions. We compute the Brown measure of the product of elliptic elements. It turns out that this Brown measure is same as the limiting spectral distribution.

On the Separability of Unitarily Invariant Random Quantum States: The Unbalanced Regime

We study entanglement-related properties of random quantum states which are unitarily invariant, in the sense that their distribution is left unchanged by conjugation with arbitrary unitary operators. In the large matrix size limit, the distribution of these random quantum states is characterized by their limiting spectrum, a compactly supported probability distribution. We prove several results characterizing entanglement and the PPT property of random bipartite unitarily invariant quantum states in terms of the limiting spectral distribution, in the unbalanced asymptotical regime where one of the two subsystems is fixed, while the other one grows in size.

Polynomial generalizations of the sample variance-covariance matrix when pn−1 → 0

Let [Formula: see text] be random matrices, where [Formula: see text] are independently distributed. Suppose [Formula: see text], [Formula: see text] are non-random matrices of order [Formula: see text] and [Formula: see text] respectively. Suppose [Formula: see text], [Formula: see text] and [Formula: see text]. Consider all [Formula: see text] random matrix polynomials constructed from the above matrices of the form [Formula: see text] [Formula: see text] and the corresponding centering polynomials [Formula: see text] [Formula: see text]. We show that under appropriate conditions on the above matrices, the variables in the non-commutative ∗-probability space [Formula: see text] with state [Formula: see text] converge. We also show that the limiting spectral distribution of [Formula: see text] exists almost surely whenever [Formula: see text] and [Formula: see text] are self-adjoint. The limit can be expressed in terms of, semi-circular, circular and other families and, limits of [Formula: see text], [Formula: see text] and non-commutative limit of [Formula: see text]. Our results fully generalize the results already known for [Formula: see text].