Using operator methods, we generally present the level densities for kinds of random matrix unitary ensembles in weak sense. As a corollary, the limit spectral distributions of random matrices from Gaussian, Laguerre and Jacobi unitary ensembles are recovered. At the same time, we study the perturbation invariability of the level densities of random matrix unitary ensembles. After the weight function associated with the 1-level correlation function is appended a polynomial multiplicative factor, the level density is invariant in the weak sense.
For a broad class of unitary ensembles of random matrices we demonstrate the universal nature of the Janossy densities of eigenvalues near the spectral edge, providing a different formulation of the probability distributions of the limiting second, third, etc. largest eigenvalues of the ensembles in question. The approach is based on a representation of the Janossy densities in terms of a system of orthogonal polynomials, plus the steepest descent method of Deift and Zhou for the asymptotic analysis of the associated Riemann-Hilbert problem.
We study the densities of limiting distributions of squared singular values of high-dimensional matrix products composed of independent complex Gaussian (complex Ginibre) and truncated unitary matrices which are taken from Haar distributed unitary matrices with appropriate dimensional growth. In the general case we develop a new approach to obtain complex integral representations for densities of measures whose Stieltjes transforms satisfy algebraic equations of a certain type. In the special cases in which at most one factor of the product is a complex Gaussian we derive elementary expression…
We generally study the density of eigenvalues in unitary ensembles of random matrices from the recurrence coefficients with regularly varying conditions for the orthogonal polynomials. First we calculate directly the moments of the density. Then, by studying some deformation of the moments, we get a family of differential equations of first order which the densities satisfy (see Theorem 1.2), and give the densities by solving them. Further, we prove that the density is invariant after the polynomial perturbation of the weight function (see Theorem 1.5).
Embedded random matrix ensembles are generic models for describing statistical properties of finite isolated interacting quantum many-particle systems. For the simplest spinless systems, with say $m$ particles in $N$ single particle states and interacting via $k$-body interactions, we have EGUE($k$) and the embedding algebra is $U(N)$. A finite quantum system, induced by a transition operator, makes transitions from its states to the states of the same system or to those of another system. Examples are electromagnetic transitions (same initial and final systems), nuclear beta and double beta d…
We propose to use eigenvalue densities of unitary random matrix ensembles as mass distributions in gravitational lensing. The corresponding lens equations reduce to algebraic equations in the complex plane which can be treated analytically. We prove that these models can be applied to describe lensing by systems of edge-on galaxies. We illustrate our analysis with the Gaussian and the quartic unitary matrix ensembles.
We use classical results from harmonic analysis on matrix spaces to investigate the relation between the joint density of the singular values and of the eigenvalues of complex random matrices which are bi-unitarily invariant (also known as isotropic or unitary rotation invariant). We prove that one of these joint densities determines the other one. Moreover we construct an explicit formula relating both joint densities at finite matrix dimension. This relation covers probability densities as well as signed densities. With the help of this relation we derive general analytical relations among t…
I use quaternion free probability calculus - an extension of free probability to non-Hermitian matrices (which is introduced in a succinct but self-contained way) - to derive in the large-size limit the mean densities of the eigenvalues and singular values of sums of independent unitary random matrices, weighted by complex numbers. In the case of CUE summands, I write them in terms of two "master equations," which I then solve and numerically test in four specific cases. I conjecture a finite-size extension of these results, exploiting the complementary error function. I prove a central limit …
Inspired by the theory of quantum information, I use two non-Hermitian random matrix models - a weighted sum of circular unitary ensembles and a product of rectangular Ginibre unitary ensembles - as building blocks of three new products of random matrices which are generalizations of the Bures model. I apply the tools of both Hermitian and non-Hermitian free probability to calculate the mean densities of their eigenvalues and singular values in the thermodynamic limit, along with their divergences at zero; the results are supported by Monte Carlo simulations. I pose and test conjectures concer…
Generation of Wigner functions of Landau levels and determination of their symmetries and generic properties are achieved in the autonomous framework of deformation quantization. Transformation properties of diagonal Wigner functions under space inversion, time reversal and parity transformations are specified and their invariance under a four-parameter subgroup of symplectic transformations are established. A generating function for all Wigner functions is developed and this has been identified as the phase-space coherent state for Landau levels. Integrated forms of generating function are us…
We study the joint probability density of the eigenvalues of a product of rectangular real, complex or quaternion random matrices in a unified way. The random matrices are distributed according to arbitrary probability densities, whose only restriction is the invariance under left and right multiplication by orthogonal, unitary or unitary symplectic matrices, respectively. We show that a product of rectangular matrices is statistically equivalent to a product of square matrices. Hereby we prove a weak commutation relation of the random matrices at finite matrix sizes, which previously have bee…
We compute individual distributions of low-lying eigenvalues of massive chiral random matrix ensembles by the Nyström-type quadrature method for evaluating the Fredholm determinant and Pfaffian that represent the analytic continuation of the Janossy densities (conditional gap probabilities). A compact formula for individual eigenvalue distributions suited for precise numerical evaluation by the Nyström-type method is obtained in an explicit form, and the $k^{\rm{\small th}}$ smallest eigenvalue distributions are numerically evaluated for chiral unitary and symplectic ensembles in the microscop…
The unitary correspondence between Quantum Hall states in higher Landau levels and states in the lowest Landau level is discussed together with the resulting transformation formulas for particle densities and interaction potentials. This correspondence leads in particular to a representation of states in arbitrary Landau levels in terms of holomorphic functions. Some important special Quantum Hall states in higher Landau levels are also briefly discussed.
We prove that QCD in the epsilon-regime of chiral Perturbation Theory is equivalent to chiral Random Matrix Theory for zero and both non-zero real and imaginary chemical potential mu. To this aim we prove a theorem that relates integrals over fermionic and bosonic variables to super-Hermitian or super-Unitary groups also called superbosonization. Our findings extend previous results for the equivalence of the partition functions, spectral densities and the quenched two-point densities. We can show that all k-point density correlation functions agree in both theories for an arbitrary number of …
The eigenvalue density for members of the Gaussian orthogonal and unitary ensembles follows the Wigner semi-circle law. If the Gaussian entries are all shifted by a constant amount c/Sqrt(2N), where N is the size of the matrix, in the large N limit a single eigenvalue will separate from the support of the Wigner semi-circle provided c > 1. In this study, using an asymptotic analysis of the secular equation for the eigenvalue condition, we compare this effect to analogous effects occurring in general variance Wishart matrices and matrices from the shifted mean chiral ensemble. We undertake an a…
For finite quantum many-particle systems, a given system, induced by a transition operator, makes transitions from its states to the states of the same system or to those of another system. Examples are electromagnetic transitions (then the initial and final systems are same), nuclear beta and double beta decay (then the initial and final systems are different), particle addition to or removal from a given system and so on. Working towards developing a complete statistical theory for transition strength densities (transition strengths multiplied by the density of states at the initial and fina…
We collect explicit and user-friendly expressions for one-point densities of the real eigenvalues $\{λ_i\}$ of $N\times N$ Wishart-Laguerre and Jacobi random matrices with orthogonal, unitary and symplectic symmetry. Using these formulae, we compute integer moments $τ_n=<\sum_{i=1}^Nλ_i^n>$ for all symmetry classes without any large $N$ approximation. In particular, our results provide exact expressions for moments of transmission eigenvalues in chaotic cavities with time-reversal or spin-flip symmetry and supporting a finite and arbitrary number of electronic channels in the two incoming lead…
We analyze the properties of entangled random pure states of a quantum system partitioned into two smaller subsystems of dimensions $N$ and $M$. Framing the problem in terms of random matrices with a fixed-trace constraint, we establish, for arbitrary $N \leq M$, a general relation between the $n$-point densities and the cross-moments of the eigenvalues of the reduced density matrix, i.e. the so-called Schmidt eigenvalues, and the analogous functionals of the eigenvalues of the Wishart-Laguerre ensemble of the random matrix theory. This allows us to derive explicit expressions for two-level de…
