Gaussian Fluctuation for the Number of Particles in Airy, Bessel, Sine, and Other Determinantal Random Point Fields

We prove the Central Limit Theorem (CLT) for the number of eigenvalues near the spectrum edge for certain Hermitian ensembles of random matrices. To derive our results, we use a general theorem, essentially due to Costin and Lebowitz, concerning the Gaussian fluctuation of the number of particles in random point fields with determinantal correlation functions. As another corollary of the Costin–Lebowitz Theorem we prove the CLT for the empirical distribution function of the eigenvalues of random matrices from classical compact groups.     

On the limit of extreme eigenvalues of large dimensional random quaternion matrices

Since E.P. Wigner (1958) established his famous semicircle law, lots of attention has been paid by physicists, probabilists and statisticians to study the asymptotic properties of the largest eigenvalues for random matrices. Bai and Yin (1988) obtained the necessary and sufficient conditions for the strong convergence of the extreme eigenvalues of a Wigner matrix. In this paper, we consider the case of quaternion self-dual Hermitian matrices. We prove the necessary and sufficient conditions for the strong convergence of extreme eigenvalues of quaternion self-dual Hermitian matrices corresponding to the Wigner case.     

Coherent noise, scale invariance and intermittency in large systems

We introduce a new class of models in which a large number of “agents” organize under the influence of an externally imposed coherent noise. The model shows reorganization events whose size distribution closely follows a power law over many decades, even in the case where the agents do not interact with each other. In addition, the system displays “aftershock” events in which large disturbances are followed by a string of others at times which are distributed according to a t⁻¹ law. We also find that the lifetimes of the agents in the system possess a power-law distribution. We explain all these results using an approximate analytic treatment of the dynamics and discuss a number of variations on the basic model relevant to the study of particular physical systems.     

Scaling laws of complex band random matrices

We study a model of complex band random matrices capable of describing the transitions between three different ensembles of Hermitian matrices: Gaussian orthogonal, Gaussian unitary and Poissonian. Analyzing numerical data we observe new scaling relations based on the generalized localization length of eigenvectors. We show that during transitions between canonical ensembles of random matrices the changes of statistical properties of eigenvalues and eigenvectors are correlated.     

Scaling laws of complex band random matrices

We study a model of complex band random matrices capable of describing the transitions between three different ensembles of Hermitian matrices: Gaussian orthogonal, Gaussian unitary and Poissonian. Analyzing numerical data we observe new scaling relations based on the generalized localization length of eigenvectors. We show that during transitions between canonical ensembles of random matrices the changes of statistical properties of eigenvalues and eigenvectors are correlated.     

The ground state of the two-leg Hubbard ladder a density-matrix renormalization group study

We present density-matrix renormalization group results for the ground state properties of two-leg Hubbard ladders. The half-filled Hubbard ladder is an insulating spin-gapped system, exhibiting a crossover from a spin liquid to a band insulator as a function of the interchain hopping matrix element. When the system is doped, there is a parameter range in which the spin gap remains. In this phase, the doped holes from singlet pairs and the pair field and the “4k_F” density correlations associated with pair-density fluctuations decay as power laws, while the “2k_F” charge density wave correlations decay exponentially. We discuss the behavior of the exponents of the pairing and density correlations within this spin-gapped phase. Additional one-band Luttinger liquid phases which occur in the large interband hopping regime are also discussed.     

The Hadron to quark-gluon transition in relativistic heavy-ion collisions

In this paper we construct a scenario for the QCD transition from the hadron phase to the quark/gluon phase using physical models for these phases. The hadron phase is modeled by a spectrum of hadrons with masses which drop (with a common scaling factor) towards zero at chiral symmetry restoration. The number of hadronic effective degrees of freedom is limited by the number of microscopic degrees of freedom in the quark/gluon phase. This limitation can be imposed either by fiat or through the introduction of a temperature-dependent excluded volume. Given that the number of degrees of freedom in hadrons and in quarks and gluons are roughly equal, the QCD phase transition is inhibited by the bag constant. The only phase transition seen in lattice-gauge calculations, once low-mass quarks are included, is the restoration of chiral symmetry which occurs at the relatively low temperature of ˜ 150 MeV. At present, lattice gauge calculations do not have the resolution to determine the properties of the higher hadronic states accurately. They do, however, demonstrate that chiral restoration takes place in the (ρ. a₁), ( +)), ( −)) and (π, σ) systems by yielding “screening masses” for chiral partners which are distinct for T < T _xSR and identical for T>T _xSR. Further, within numerical accuracy, these “screening masses” are consistent with pure thermal energies and show no evidence of remaining bare masses once chiral symmetry is restored. These, and other lattice-gauge results, will be discussed in the light of our scenario. We shall also consider the consequences of our picture for relativistic heavy-ion experiments.     

Chiral symmetry and the A1 contribution to the nucleon-nucleon interaction

The contribution of A₁ exchange to the nucleon-nucleon potential is studied in a broken chiral symmetric model. The A₁ is treated as a finite-width resonance in the πρ s-wave. Connections between pseudoscalar and pseudovector pion-nucleon coupling in the underlying model lagrangian are studied in detail. It is found that large terms in the NN interaction arising from πρ exchange with pseudoscalar coupling are suppressed by interference with a₁ exchange. With pseudovector coupling there is a suppression of the A₁ exchange by the so-called “seagull” terms in πρ exchange which arise from gauge invariance. The suppression becomes an exact cancellation in the limit of infinite ρ and a₁ masses and exact chiral symmetry. We found that inclusion of the a₁ decay into the πρ state strongly modifies the a₁] exchange potential, suppressing the tensor part but leaving the spin-spin part almost unchanged.     

Characteristic polynomials of complex random matrix models

We calculate the expectation value of an arbitrary product of characteristic polynomials of complex random matrices and their Hermitian conjugates. Using the technique of orthogonal polynomials in the complex plane our result can be written in terms of a determinant containing these polynomials and their kernel. It generalizes the known expression for Hermitian matrices and it also provides a generalization of the Christoffel formula to the complex plane. The derivation we present holds for complex matrix models with a general weight function at finite-N, where N is the size of the matrix. We give some explicit examples at finite-N for specific weight functions. The characteristic polynomials in the large-N limit at weak and strong non-hermiticity follow easily and they are universal in the weak limit. We also comment on the issue of the BMN large-N limit.     

On the Law of Multiplication of Random Matrices

We recover Voiculescu's results on multiplicative free convolutions of probability measures by different techniques which were already developed by Pastur and Vasilchuk for the law of addition of random matrices. Namely, we study the normalized eigenvalue counting measure of the product of two n×n unitary matrices and the measure of the product of three n×n Hermitian (or real symmetric) positive matrices rotated independently by random unitary (or orthogonal) Haar distributed matrices. We establish the convergence in probability as n to a limiting nonrandom measure and obtain functional equations for the Herglotz and Stieltjes transforms of that limiting measure.     

On the reconstruction of the conduction electron spectrum in metal-oxide superconductors owing to the interaction with coherent atomic displacements

A “pseudospin ferromagnet” model is proposed which describes the interaction of conduction electrons with coherent atomic displacements in metal-oxide high-T_c superconductors (HTSC). Non-quasiparticle states (“pseudospinons”) play an important role in this model. They make an appreciable contribution to thermodynamic and transport properties (e.g., to the linear term in specific heat), although not contributing to the density of states at E_F at T=0. In the superconducting phase, the pseudospinons give rise to gapless superconductivity at finite temperatures. For certain model parameter relations, a new energy scale (“Kondo temperature”) may occur in the electron spectrum near E_F. Using the results obtained, experimental data on the thermopower, nuclear spin-lattice relaxation rate and other properties of HTSC are discussed.     

Laplace operator and random walk on one-dimensional nonhomogeneous lattice

A classical result of probability theory states that under suitable space and time renormalization, a random walk converges to Brownian motion. We prove an analogous result in the case of nonhomogeneous random walk on onedimensional lattice. Under suitable conditions on the nonhomogeneous medium, we prove convergence to Brownian motion and explicitly compute the diffusion coefficient. The proofs are based on the study of the spectrum of random matrices of increasing dimension.     

Order parameters and boundary effects in U(1) lattice gauge theory

We show that, independently of the boundary conditions, the two phases of the 4-dimensional compact U(1) lattice gauge theory can be characterized by the presence or absence of an “infinite” current network, with an appropriate definition of “infinite” network takes values 0 or 1 in the cold and hot phase, respectively. It thus constitutes a very efficient order parameter, which allows one to determine the transition region at low computational cost. In addition, for open and fixed boundary conditions we address the question of the impact of inhomogeneities and give examples of the reappearance of an energy gap already at moderate lattice sizes.     

Approximate solution of the Schrödinger equation for an atom in the field of a standing wave

We present an approximate method of calculation of energy spectrum and wave functions of translational motion of a two-level atom in the field of a standing wave at zero value of the quasimomentum. Wave functions in the momentum representation are expressed in terms of confluent hypergeometric functions in case of continuous spectrum and in terms of Hermitian polynomials in case of discrete spectrum. The energy spectrum exhibits infinite compaction at transition from discrete part to continuous, as it is the case for, e.g., Coulombian potential.     

Exact Wigner Surmise Type Evaluation of the Spacing Distribution in the Bulk of the Scaled Random Matrix Ensembles

Random matrix ensembles with orthogonal and unitary symmetry correspond to the cases of real symmetric and Hermitian random matrices, respectively. We show that the probability density function for the corresponding spacings between consecutive eigenvalues can be written exactly in the Wigner surmise type form a(s)e^–b(s) for a simply related to a Painlevé transcendent and b its anti-derivative. A formula consisting of the sum of two such terms is given for the symplectic case (Hermitian matrices with real quaternion elements).     

Spectral Analysis of Unitary Band Matrices

 This paper is devoted to the spectral properties of a class of unitary operators with a matrix representation displaying a band structure. Such band matrices appear as monodromy operators in the study of certain quantum dynamical systems. These doubly infinite matrices essentially depend on an infinite sequence of phases which govern their spectral properties. We prove the spectrum is purely singular for random phases and purely absolutely continuous in case they provide the doubly infinite matrix with a periodic structure in the diagonal direction. We also study some properties of the singular spectrum of such matrices considered as infinite in one direction only. Received: 29 April 2002 / Accepted: 7 August 2002 Published online: 20 January 2003 Communicated by B. Simon     

Vortices in weakly coupled layered superconductors

The vortex system in high-temperature layered superconductors exhibits a rich phase diagram with many proposals of phase transitions modifying the correlations both within and between the layers. We focus on the limit where the magnetic coupling between “pancake” vortices dominates over the interlayer Josephson coupling. The weak, but long-ranged nature of this magnetic interaction allows for an accurate “mean-field” treatment where the pancakes in each layer move independently in a self-consistent substrate potential. We calculate the form of the two relevant phase transitions in this system. First, we determine when the substrate potential is too weak to stabilize the two-dimensional (2D) fluctuations and the lattice evaporates to a pancake gas. Second, within the lattice we find a Kosterlitz–Thouless unbinding transition of vacancies and interstitials. For a small but finite Josephson term, this is identified with the phase-decoupling transition.     

Evaporation of droplets in the two-dimensional Ginzburg–Landau equation

We consider the problem of coarsening in two dimensions for the real (scalar) Ginzburg–Landau equation. This equation has exactly two stable stationary solutions, the constant functions +1 and −1. We assume most of the initial condition is in the “−1” phase with islands of “+1” phase. We use invariant manifold techniques to prove that the boundary of a circular island moves according to Allen–Cahn curvature motion law. We give a criterion for non-interaction of two arbitrary interfaces and a criterion for merging of two nearby interfaces.