Related papers: The Hermitian two matrix model with an even quarti…
We consider the random matrix model with external source, in case where the potential V(x) is an even polynomial and the external source has two eigenvalues a, -a of equal multiplicity. We show that the limiting mean eigenvalue distribution…
In this paper we studied the asymptotic eigenvalue statistics of the 2 matrix model with a quartic monomial and a general even polynomial potential. We studied the correlation kernel for the eigenvalues of one of the matrices in asymptotic…
We study Hermitian random matrix models with an external source matrix which has equispaced eigenvalues, and with an external field such that the limiting mean density of eigenvalues is supported on a single interval as the dimension tends…
The eigenvalue statistics of a pair $(M_1,M_2)$ of $n\times n$ Hermitian matrices taken random with respect to the measure $$\frac{1}{Z_n}\exp\big(-n\Tr (V(M_1)+W(M_2)-\tau M_1M_2)\big) {\rm d}M_1 {\rm d} M_2 $$ can be described in terms of…
After reviewing the Hermitian one matrix model, we will give a brief introduction to the Hermitian two matrix model and present a summary of some recent results on the asymptotic behavior of the two matrix model with a quartic potential. In…
In this paper, we derive new relative perturbation bounds for eigenvectors and eigenvalues for regular quadratic eigenvalue problems of the form $\lambda^2 M x + \lambda C x + K x = 0$, where $M$ and $K$ are nonsingular Hermitian matrices…
In this thesis generalizations of matrix and eigenvalue models involving supersymmetry are discussed. Following a brief review of the Hermitian one matrix model, the c=-2 matrix model is considered. Built from a matrix valued superfield…
For a pair of coupled rectangular random matrices we consider the squared singular values of their product, which form a determinantal point process. We show that the limiting mean distribution of these squared singular values is described…
We consider the random Hermitian matrix model of dimension $2n$, with external source, defined by the probability density function \begin{equation*} \frac{1}{Z_{2n}} \lvert \det(M) \rvert^{\alpha} e^{-2n\mathrm{Tr} (V(M) - AM)}, \quad V(x)…
We establish lower semi-continuity and strict convexity of the energy functionals for a large class of vector equilibrium problems in logarithmic potential theory. This in particular implies the existence and uniqueness of a minimizer for…
We consider the hermitian random matrix model with external source and general polynomial potential, when the source has two distinct eigenvalues but is otherwise arbitrary. All such models studied so far have a common feature: an…
We study the hermitian one matrix model with semi-classical potential. This is a general unitary invariant random matrix ensemble in which the potential has a derivative that is a rational function and the measure is supported on some…
We consider a notion of balanced metrics for triples (X,L,E) which depend on a parameter \alpha, where X is smooth complex manifold with an ample line bundle L and E is a holomorphic vector bundle over X. For generic choice of \alpha, we…
The general expression with the physical significance and positive definite condition of the eigenvalues of $4\times 4$ Hermitian and trace-one matrix are obtained. This implies that the eigenvalue problem of the $4\times 4$ density matrix…
The study deals with a minimal energy problem in the presence of an external field over noncompact classes of vector measures of infinite dimension in a locally compact space. The components are positive measures (charges) satisfying…
We consider the random matrix ensemble with an external source \[ \frac{1}{Z_n} e^{-n \Tr({1/2}M^2 -AM)} dM \] defined on $n\times n$ Hermitian matrices, where $A$ is a diagonal matrix with only two eigenvalues $\pm a$ of equal…
A two dimensional eigenvalue problem (2DEVP) of a Hermitian matrix pair $(A, C)$ is introduced in this paper. The 2DEVP can be viewed as a linear algebraic formulation of the well-known eigenvalue optimization problem of the parameter…
This paper is concerned with two extremal problems from matrix analysis. One is about approximating the top eigenspaces of a Hermitian matrix and the other one about approximating the orthonormal polar factor of a general matrix. Tight…
We construct a previously unknown $E_2$-quasi-exactly solvable non-Hermitian model whose eigenfunctions involve weakly orthogonal polynomials obeying three-term recurrence relations that factorize beyond the quantization level. The model…
We continue the study of the Hermitian random matrix ensemble with external source $\frac{1}{Z_n} e^{-n \Tr({1/2}M^2 -AM)} dM$ where $A$ has two distinct eigenvalues $\pm a$ of equal multiplicity. This model exhibits a phase transition for…