Random Unitary Matrices, Permutations and Painlevé

This paper is concerned with certain connections between the ensemble of n×n unitary matrices – specifically the characteristic function of the random variable tr(U) – and combinatorics – specifically Ulam's problem concerning the distribution of the length of the longest increasing subsequence in permutation groups – and the appearance of Painlevé functions in the answers to apparently unrelated questions. Among the results is a representation in terms of a Painlevé V function for the characteristic function of tr(U) and (using recent results of Baik, Deift and Johansson) an expression in terms of a Painlevé II function for the limiting distribution of the length of the longest increasing subsequence in the hyperoctahedral groups. Received: 2 December 1998 / Accepted: 12 May 1999     

Einstein—Podolsky—Rosen Correlations and the Preferred Frame

The correlation function of spin measurements of two spin- particles in two moving inertial frames is derived within the framework of the Lorentz covariant quantum mechanics with the preferred frame. The localization of the particles during the detection and proper transformation properties under the action of the Lorentz group of the spin operator are taken into account. Some special cases and approximations of the calculated correlation function are discussed.     

Distributions on Partitions, Point Processes,¶ and the Hypergeometric Kernel

We study a 3-parametric family of stochastic point processes on the one-dimensional lattice originated from a remarkable family of representations of the infinite symmetric group. We prove that the correlation functions of the processes are given by determinantal formulas with a certain kernel. The kernel can be expressed through the Gauss hypergeometric function; we call it the hypergeometric kernel. In a scaling limit our processes approximate the processes describing the decomposition of representations mentioned above into irreducibles. As we showed in previous works, the correlation functions of these limit processes also have determinantal form with so-called Whittaker kernel. We show that the scaling limit of the hypergeometric kernel is the Whittaker kernel. integrable operator as defined by Its, Izergin, Korepin, and Slavnov. We argue that the hypergeometric kernel can be considered as a kernel defining a ‘discrete integrable operator’. We also show that the hypergeometric kernel degenerates for certain values of parameters to the Christoffel–Darboux kernel for Meixner orthogonal polynomials. This fact is parallel to the degeneration of the Whittaker kernel to the Christoffel–Darboux kernel for Laguerre polynomials. Received: 22 September 1999 / Accepted: 23 November 1999     

Resolving the Bethe–Salpeter Kernel

A novel method for constructing a kernel for the meson bound-state problem is described.It produces a closed form that is symmetry-consistent(discrete and continuous) with the gap equation defined by any admissible gluon-quark vertex,Γ.Applicable even when the diagrammatic content of Γ is unknown,the scheme can foster new synergies between continuum and lattice approaches to strong interactions.The framework is illustrated by showing that the presence of a dressed-quark anomalous magnetic moment in Γ,an emergent feature of strong interactions,can remedy many defects of widely used meson bound-state kernels,including the mass splittings between vector and axial-vector mesons and the level ordering of pseudoscalar and vector meson radial excitations.     

What Do These Correlations Know about Reality? Nonlocality and the Absurd

In honor of Daniel Greenberger's 65th birthday, I record for posterity two superb examples of his wit, offer a proof of an important theorem on quantum correlations that even those of us over 60 can understand, and suggest, by trying to make it look silly, that invoking quantum nonlocality as an explanation for such correlations may be too cheap a way out of the dilemma they pose.     

Exact Calculations of Vertex ^-sγb and ,^-sZb in the Unitary Gauge

In this paper, we present the exact calculations for the vertex ^-sγb and ^sZb in the unitary gauge. We find that （a） the divergent- and μ-dependent terms are left in the effective vertex function Г^γμ（p, k） for b → sγ transition even after we sum up the contributions from four related Feynman diagrams; （b） for an on-shell photon, such terms do not contribute et al.; （c） for off-shell photon, these terms will be canceled when the contributions from both vertex ^sγb and ^sZb are taken into account simultaneously, and therefore the finite and gauge-independent function Zo（xt） = Co（xt） ＋ Do（xt）/4, which governs the semi-leptonic decay b → sl^- l^＋, is derived in the unitary gauge.     

Bell’s Correlations and Spin Systems

The structure of maximal violators of Bell’s inequalities for Jordan algebras is investigated. It is proved that the spin factor V ₂ is responsible for maximal values of Bell’s correlations in a faithful state. In this situation maximally correlated subsystems must overlap in a nonassociative subalgebra. For operator commuting subalgebras it is shown that maximal violators have the structure of the spin systems and that the global state (faithful on local subalgebras) acts as the trace on local subalgebras.     

Energy Correlations in O(N) Models¶and the Wolff Representation

We prove that energy functions are positively correlated in isotropic, ferromagnetic O(N) models on an arbitrary graph. In our inductive proof, this is used to prove the strong FKG property of the Wolff representation for isotropic, ferromagnetic O(N+ 1) models. This strong FKG property is then used to prove energy correlations for the O(N+ 1) model. Furthermore, percolation in the Wolff representation is proved to be a necessary and sufficient condition for positivity of the spontaneous magnetization (previously known only for N= 3). Received: 7 March 2000 / Accepted: 31 October 2000     

On the Characteristic Polynomial¶ of a Random Unitary Matrix

We present a range of fluctuation and large deviations results for the logarithm of the characteristic polynomial Z of a random N×N unitary matrix, as N→∞. First we show that , evaluated at a finite set of distinct points, is asymptotically a collection of i.i.d. complex normal random variables. This leads to a refinement of a recent central limit theorem due to Keating and Snaith, and also explains the covariance structure of the eigenvalue counting function. Next we obtain a central limit theorem for ln Z in a Sobolev space of generalised functions on the unit circle. In this limiting regime, lower-order terms which reflect the global covariance structure are no longer negligible and feature in the covariance structure of the limiting Gaussian measure. Large deviations results for ln Z/A, evaluated at a finite set of distinct points, can be obtained for . For higher-order scalings we obtain large deviations results for ln Z/A evaluated at a single point. There is a phase transition at A= ln N (which only applies to negative deviations of the real part) reflecting a switch from global to local conspiracy. Received: 27 June 2000 / Accepted: 30 January 2001     

Unitary fermions and Lscher's formula on a crystal

We consider the low-energy particle-particle scattering properties in a periodic simple cubic crystal. In particular, we investigate the relation between the two-body scattering length and the energy shift experienced by the lowest-lying unbound state when this is placed in a periodic finite box. We introduce a continuum model for s-wave contact interactions that respects the symmetry of the Brillouin zone in its regularisation and renormalisation procedures, and corresponds to the nae continuum limit of the Hubbard model. The energy shifts are found to be identical to those obtained in the usual spherically symmetric renormalisation scheme upon resolving an important subtlety regarding the cutoff procedure. We then particularize to the Hubbard model, and find that for large finite lattices the results are identical to those obtained in the continuum limit. The results reported here are valid in the weak,intermediate and unitary limits. These may be used to significantly ease the extraction of scattering information, and therefore effective interactions in condensed matter systems in realistic periodic potentials. This can achieved via exact diagonalisation or Monte Carlo methods, without the need to solve challenging, genuine multichannel collisional problems with very restricted symmetry simplifications.     

Pregelation Behaviour of Coagulation Processes with the Constant-Reaction-Number Kernel

We propose an irreversible binary coagulation model with a constant-reaction-number kernel, in which, among all the possible binary coagulation reactions, only p reactions are permitted to take place at every time. By means of the generalized rate equation we investigate the kinetic behaviour of the system with the reaction rate kernel K（i;j） = （ij）^w （0 ≤w〈1/2）, at which an i-mer and a j-mer coagulate together to form a large one. It is found that for such a system there always exists a gelation transition at a finte time to, which is in contrast to the ordinary binary coagulation with the same rate kernel. Moreover, the pre-gelation behaviour of the cluster size distribution near the gelation point falls in a scaling regime and the typical cluster size grows as （to - t）-1/（1-2w）. On the other hand, our model can also provide some predictions for the evolution of the cluster distribution in multicomponent complex networks.     

Continuous and Discrete Painlevé Equations Arising from the Gap Probability Distribution of the Finite n Gaussian Unitary Ensembles

In this paper we study the gap probability problem in the Gaussian unitary ensembles of \(n\) by \(n\) matrices : The probability that the interval \(J := (-a,a)\) is free of eigenvalues. In the works of Tracy and Widom, Adler and Van Moerbeke, and Forrester and Witte on this subject, it has been shown that two Painlevé type differential equations arise in this context. The first is the Jimbo–Miwa–Okomoto \(\sigma \) -form and the second is a particular Painlevé IV. Using the ladder operator technique of orthogonal polynomials we derive three quantities associated with the gap probability, denoted as \(\sigma _n(a)\) , \(R_n(a)\) and \(r_n(a)\) . We show that each one satisfies a second order Painlevé type differential equation as well as a discrete Painlevé type equation. In particular, in addition to providing an elementary derivation of the aforementioned \(\sigma \) -form and Painlevé IV we are able to show that the quantity \(r_n(a)\) satisfies a particular case of Chazy’s second degree second order differential equation. For the discrete equations we show that the quantity \(r_n(a)\) satisfies a particular form of the modified discrete Painlevé II equation obtained by Grammaticos and Ramani in the context of Backlund transformations. We also derive second order second degree difference equations for the quantities \(R_n(a)\) and \(\sigma _n(a)\) .     

The vector coupling α V((r) and the scales r 0, r 1 in the background perturbation theory

We study the universal static potential V _st(r) and the force, which are fully determined by two fundamental parameters: the string tension σ = 0.18 ± 0.02 GeV² and the QCD constants \(\Lambda _{\overline {MS} } (n_f )\) , taken from pQCD, while the infrared (IR) regulator M _B is taken from the background perturbation theory and expressed via the string tension. The vector couplings α _V(r) in the static potential and α _F(r) in the static force, as well as the characteristic scales, r ₁(n _f = 3) and r ₀(n _f = 3), are calculated and compared to lattice data. The result \(r_0 \Lambda _{\overline {MS} } (n_f = 3) = 0.77 \pm 0.03\) , which agrees with the lattice data, is obtained for M _B = (1.15 ± 0.02) GeV. However, better agreement with the bottomonium spectrum is reached for a smaller \(\Lambda _{\overline {MS} } (n_f = 3) = (325 \pm 15)\) MeV and the frozen value of α _V = 0.57 ± 0.02. The mass splittings \(\bar M(1D) - \bar M(1P)\) and \(\bar M(2P) - \bar M(1P)\) are shown to be sensitive to the IR regulator used. The masses M(1 ³ D ₃) = 10169(2) MeV andM(1 ³ D ₁) = 10155(3) MeV are predicted.     

Correlations between the Interacting Fragments of Decaying Processes

We study the correlations (and alignment as a particular case) existent between the fragments originated in a decaying process when the daughter particles interact. The interaction between the particles is modeled using the potential of coupled oscillators, which can be treated analytically. This approach can be considered as a first step towards the characterization of realistic interacting decaying systems, an archetypal process in physics. The results presented here also suggest the possibility of manipulating correlations using external fields, a technique that could be useful to provide sources of entangled massive particles.     

Study of the asymptotic Cramér–Rao Bound for the COLD uniform linear array

In this paper, we study the Cocentered Orthogonal Loop and Dipole pairs Uniform Linear Array (COLD-ULA) which is sensitive to the source polarization in the context of the localization of time-varying narrow-band far-field sources. We derive and analyze nonmatrix expressions of the deterministic Cramér–Rao Bound (${\text{CRB}}^{\mathrm{(COLD)}}$) for the direction and the polarization parameters under the assumption that all the sources are lying in the azimuthal plane. We denote this bound by ${\text{ACRB}}^{\mathrm{(COLD)}}$, where the “A” stands for Asymptotic, meaning that the presented results are derived under the assumption that the number of sensors is sufficiently large. While, to our knowledge, closed-form (nonmatrix) expressions of the ${\text{CRB}}^{\mathrm{(COLD)}}$ for multiple time-varying polarized sources signal do not exist in the literature, we show that the ${\text{ACRB}}^{\mathrm{(COLD)}}$ takes a closed-form (nonmatrix) expression in this context and is a good approximation of the ${\text{CRB}}^{\mathrm{(COLD)}}$ even if the number of sensor is moderate (about ten), if the source signals are not spatially too close. Our approach has two important advantages: (i) the computational complexity of the proposed closed-form of the bound is very low, compared to the brute force computation of a matrix-based deterministic CRB in case of time-varying model parameters and (ii) useful informations can be deduced from the closed-form expression on the behavior of the bound. In particular, we prove that the ${\text{ACRB}}^{\mathrm{(COLD)}}$ for the direction parameter is not affected by the knowledge or the lack of it concerning the polarization parameters. Another conclusion is that with a COLD-ULA, more model parameters can be estimated than for the uniformly polarized ULA without degrading the estimation accuracy of the localization parameter. Finally, we also study the ${\text{ACRB}}^{\mathrm{(COLD)}}$ for a priori known complex amplitudes.     

Quantum Perfect Correlations and Hardy’s Nonlocality Theorem

In this paper the failure of Hardy's nonlocality proof for the class of maximally entangled states is considered. A detailed analysis shows that the incompatibility of the Hardy equations for this class of states physically originates from the fact that the existence of quantum perfect correlations for the three pairs of two-valued observables (D ₁₁, D ₂₁), (D ₁₁, D ₂₂), and (D ₁₂, D ₂₁) [in the sense of having with certainty equal (different) readings for a joint measurement of any one of the pairs (D ₁₁, D ₂₁), (D ₁₁, D ₂₂), and (D ₁₂, D ₂₁)], necessarily entails perfect correlation for the pair of observables (D ₁₂, D ₂₂) [in the sense of having with certainty equal (different) readings for a joint measurement of the pair (D ₁₂, D ₂₂)]. Indeed, the set of these four perfect correlations is found to satisfy the CHSH inequality, and then no violations of local realism will arise for the maximally entangled state as far as the four observables D _ij, i,j = 1 or 2, are concerned. The connection between this fact and the impossibility for the quantum mechanical predictions to give the maximum possible theoretical violation of the CHSH inequality is pointed out. Moreover, it is generally proved that the fulfillment of all the Hardy nonlocality conditions necessarily entails a violation of the resulting CHSH inequality. The largest violation of this latter inequality is determined.