Matrices coupled in a chain. I. Eigenvalue correlations

The general correlation function for the eigenvalues of $p$ complex hermitian matrices coupled in a chain is given as a single determinant. For this we use a slight generalization of a theorem of Dyson.Comment: ftex eynmeh.tex, 2 files, 8 pages Submitted to: J. Phys.

Probability density of determinants of random matrices

In this brief paper the probability density of a random real, complex and quaternion determinant is rederived using singular values. The behaviour of suitably rescaled random determinants is studied in the limit of infinite order of the matrices

Moments of the characteristic polynomial in the three ensembles of random matrices

Moments of the characteristic polynomial of a random matrix taken from any of the three ensembles, orthogonal, unitary or symplectic, are given either as a determinant or a pfaffian or as a sum of determinants. For gaussian ensembles comparing the two expressions of the same moment one gets two remarkable identities, one between an $n\times n$ determinant and an $m\times m$ determinant and another between the pfaffian of a $2n\times 2n$ anti-symmetric matrix and a sum of $m\times m$ determinants.Comment: tex, 1 file, 15 pages [SPhT-T01/016], published J. Phys. A: Math. Gen. 34 (2001) 1-1

Density-functional theory for fermions in the unitary regime

In the unitary regime, fermions interact strongly via two-body potentials that exhibit a zero range and a (negative) infinite scattering length. The energy density is proportional to the free Fermi gas with a proportionality constant $\xi$. We use a simple density functional parametrized by an effective mass and the universal constant $\xi$, and employ Kohn-Sham density-functional theory to obtain the parameters from fit to one exactly solvable two-body problem. This yields $\xi=0.42$ and a rather large effective mass. Our approach is checked by similar Kohn-Sham calculations for the exactly solvable Calogero model.Comment: 5 pages, 2 figure

Using the Δ3\Delta_3 statistic to test for missed levels in mixed sequence neutron resonance data

The $\Delta_3(L)$ statistic is studied as a tool to detect missing levels in the neutron resonance data where 2 sequences are present. These systems are problematic because there is no level repulsion, and the resonances can be too close to resolve. $\Delta_3(L)$ is a measure of the fluctuations in the number of levels in an interval of length $L$ on the energy axis. The method used is tested on ensembles of mixed Gaussian Orthogonal Ensemble (GOE) spectra, with a known fraction of levels ($x$) randomly depleted, and can accurately return $x$. The accuracy of the method as a function of spectrum size is established. The method is used on neutron resonance data for 11 isotopes with either s-wave neutrons on odd-A, or p-wave neutrons on even-A. The method compares favorably with a maximum likelihood method applied to the level spacing distribution. Nuclear Data Ensembles were made from 20 isotopes in total, and their $\Delta_3(L)$ statistic are discussed in the context of Random Matrix Theory.Comment: 22 pages, 12 figures, 4 table

Calculation of some determinants using the s-shifted factorial

Several determinants with gamma functions as elements are evaluated. This kind of determinants are encountered in the computation of the probability density of the determinant of random matrices. The s-shifted factorial is defined as a generalization for non-negative integers of the power function, the rising factorial (or Pochammer's symbol) and the falling factorial. It is a special case of polynomial sequence of the binomial type studied in combinatorics theory. In terms of the gamma function, an extension is defined for negative integers and even complex values. Properties, mainly composition laws and binomial formulae, are given. They are used to evaluate families of generalized Vandermonde determinants with s-shifted factorials as elements, instead of power functions.Comment: 25 pages; added section 5 for some examples of application

Finite-difference distributions for the Ginibre ensemble

The Ginibre ensemble of complex random matrices is studied. The complex valued random variable of second difference of complex energy levels is defined. For the N=3 dimensional ensemble are calculated distributions of second difference, of real and imaginary parts of second difference, as well as of its radius and of its argument (angle). For the generic N-dimensional Ginibre ensemble an exact analytical formula for second difference's distribution is derived. The comparison with real valued random variable of second difference of adjacent real valued energy levels for Gaussian orthogonal, unitary, and symplectic, ensemble of random matrices as well as for Poisson ensemble is provided.Comment: 8 pages, a number of small changes in the tex

The Local Semicircle Law for Random Matrices with a Fourfold Symmetry

We consider real symmetric and complex Hermitian random matrices with the additional symmetry $h_{xy}=h_{N-x,N-y}$. The matrix elements are independent (up to the fourfold symmetry) and not necessarily identically distributed. This ensemble naturally arises as the Fourier transform of a Gaussian orthogonal ensemble (GOE). It also occurs as the flip matrix model - an approximation of the two-dimensional Anderson model at small disorder. We show that the density of states converges to the Wigner semicircle law despite the new symmetry type. We also prove the local version of the semicircle law on the optimal scale.Comment: 20 pages, to appear in J. Math. Phy

Initial-state randomness as a universal source of decoherence

We study time evolution of entanglement between two qubits, which are part of a larger system, after starting from a random initial product state. We show that, due to randomness in the initial product state, entanglement is present only between directly coupled qubits and only for short times. Time dependence of the entanglement appears essentially independent of the specific hamiltonian used for time evolution and is well reproduced by a parameter-free two-body random matrix model.Comment: 8 pages, 6 figure

Wigner surmise for Hermitian and non-Hermitian Chiral random matrices

We use the idea of a Wigner surmise to compute approximate distributions of the first eigenvalue in chiral Random Matrix Theory, for both real and complex eigenvalues. Testing against known results for zero and maximal non-Hermiticity in the microscopic large-N limit we find an excellent agreement, valid for a small number of exact zero-eigenvalues. New compact expressions are derived for real eigenvalues in the orthogonal and symplectic classes, and at intermediate non-Hermiticity for the unitary and symplectic classes. Such individual Dirac eigenvalue distributions are a useful tool in Lattice Gauge Theory and we illustrate this by showing that our new results can describe data from two-colour QCD simulations with chemical potential in the symplectic class