On the critical exponent for random walk intersections

The exponent _d for the probability of nonintersection of two random walks starting at the same point is considered. It is proved that 1/2<₂3/4. Monte Carlo simulations are done to suggest ₂=0.61 and ₃0.29.     

Radiative decays of the Z boson

We calculate the Stokes parameters of the photons produced in the decays of neutral vector bosons Z, Z 1+¯1+ and Z q+¯q+, wherel=e, , or , and q is a quark. In the decays of unpolarized Z bosons (with the production of unpolarized leptons or quarks) the nonzero Stokes parameters for ₂ (circularly polarized photons) and ₃ (linearly polarized photons). The magnitude of ₃ does not depend on the parameters of the netural weak current of the leptons and the quarks (if their mass is neglected). The anomalous magnetic moment of the Z boson is studied.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 38–43, February, 1986.     

Renormalization group equations with several length scales and crossover scaling functions for axial anisotropy

The critical behaviour of axially anisotropicn-vector models is characterized by two distinct length scales, the correlation lengths and for the easy and hard axes. In order to handle the full range of anisotropics from to partial differential renormalization group equations are derived, depending on and . The anisotropicX-Y model is studied in detail near four dimensions. The crossover scaling functions for the susceptibilities are calculated to first order in=4–d. Two distinct crossover regions are found for weak and dominant anisotropy, respectively.     

Radiation of relativistic electrons in a magnetic undulator

Expressions are obtained for the spectral-angular characteristics of the radiation in two limiting cases: 1 and 1 ( is the angle of deflection of the electron in the field, and is the energy of the electron in units of mc²). It is shown that in the latter case the maximum of the radiation occurs at higher harmonics.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 88–91, October, 1973.In conclusion the authors thank Professor A. A. Sokolov for useful discussions.     

Correlated majority model

We extend the bichromatic majority model by including (one-dimensional isotropic) correlations and numerically discuss, through Monte Carlo simulations, the simple, 1/3, and 2/3 majority rules. We calculate, as functions of the concentration and correlation degree, the mean finite cluster size, and the order parameterm as well as their respective critical exponents and. We find1 regardless of the correlation degree or the type of majority. Also, a supplementary divergence of is observed at the>0 borderline.     

Fluctuating hydrodynamics and principal oscillation pattern analysis

Principal oscillation pattern (POP) analysis was recently introduced into climatology to analyze multivariate time series x_i(t) produced by systems whose dynamics are described by a linear Markov process x=Bx + . The matrixB gives the deterministic feedback and is a white noise vector with covariances (t) _j (t*Q _ij (t–t. The POP method is applied to data from a direct simulation Monte Carlo program. The system is a dilute gas with 50,000 particles in a Rayleigh-Bénard configuration. The POP analysis correctly reproduces the linearized Navier-Stokes equations (in the matrixB) and the stochastic fluxes (in the matrixQ) as given by Landau-Lifschitz fluctuating hydrodynamics. Using this method, we find the Landau-Lifschitz theory to be valid both in equilibrium and near the critical point of Rayleigh-Bénard convection.     

On the dynamics of excitations in disordered systems

A new, time-local (TL) reduced equation of motion for the probability distribution of excitations in a disordered system is developed. ToO(k²) the TL equation results in a Gaussian spatial probability distribution, i.e, P(r, t) = [(2)^1/2]^–dexp(-r²/2²), where = (t) is a correlation length, andr = ¦r¦. The corresponding distribution derived from the Hahn-Zwanzig (HZ) equation is more complicated and assumes the asymptotic (r ) form: P(r, s)(s ^d )^–1exp(–r/) · (r/)^(1-d)/2 where = (s),d is the space dimensionality, ands is the Laplace transform variable conjugate tot. The HZ distribution generalizes the scaling form suggested by Alexanderet al. ford= 1. In the Markov limit (t)t, (s)1/s, and the two distributions are identical (ordinary diffusion).     

High frequency gravitational radiation in Kerr-Schild space-times

Vaidya has obtained general solutions of the Einstein equationsR _ab= _{a b} by means of the Kerr-Schild metricsg _ab= _ab +H _{a b} . The vector field _a generates a shear free null geodetic congruence both in Minkowski space and in the Kerr-Schild space-time. If in addition it is hypersurface orthogonal, the Kerr-Schild metric may be interpreted as the background metric in a space-time perturbed by a high frequency gravitational wave. It is shown that Vaidya's solutions satisfying this additional condition are of only two types: (1) Kinnersley's accelerating point mass solution and (2) a similar solution where a space-like curve plays the role of the time-like curve describing the world line of the accelerating mass. The solution named by Vaidya as the radiating Kerr metric does not satisfy the hypersurface orthogonal condition.Supported in part by National Science Foundation Grant MPS 741029.     

Nonlocal regularization and spontaneously broken abelian gauge theories for an arbitrary gauge parameter

We study the nonlocal regularization for the case of a spontaneously broken abelian gauge theory in the R-gauge with an arbitrary gauge parameter . We consider a simple abelian-Higgs model with chiral couplings as an example. We show that if we apply the nonlocal regularization procedure (to construct a nonlocal theory with FINITE mass parameter) to the spontaneously broken R-gauge Lagrangian, using the quadratic forms as appearing in this Lagrangian, we find that a physical observable in this model, an analogue of the muon anomalous magnetic moment, evaluated to order O [g²] does indeed show -dependence. We then apply the modified form of nonlocal regularization that was recently advanced and studied for the unbroken non-abelian gauge theories and discuss the resulting WT identities and -independence of the S-matrix elements.     

Distribution of roots of random real generalized polynomials

The average density of zeros for monic generalized polynomials, , with real holomorphic ,f _k and real Gaussian coefficients is expressed in terms of correlation functions of the values of the polynomial and its derivative. We obtain compact expressions for both the regular component (generated by the complex roots) and the singular one (real roots) of the average density of roots. The density of the regular component goes to zero in the vicinity of the real axis like |lmz|. We present the low- and high-disorder asymptotic behaviors. Then we particularize to the large-n limit of the average density of complex roots of monic algebraic polynomials of the form with real independent, identically distributed Gaussian coefficients having zero mean and dispersion . The average density tends to a simple,universal function of =2nlog|z| and in the domain coth(/2)n|sin arg(z)|, where nearly all the roots are located for largen.     

The Countable Number of Models in the Cosmology with Nonminimal Coupling and Torsion

Cosmological models of flat space with a nonminimally coupled scalar field and ultrarelativistic gas are studied within the Einstein–Kartan theory. Exact general solutions are derived for two-component models and those containing only scalar field for an arbitrary coupling constant . It is shown that both singular and countable number of nonsingular models is possible depending on the type of scalar field and the sign of . The special values of and restrictions on are found for the above solutions. The role of relativistic gas in the evolution of models is revealed.     

Multiatom interactions,order and stability in binary transition metal alloys

Interface delocalization or depinning transitions such as wetting or surface induced disorder are considered. At these transitions, the correlation length for transverse correlations parallel to the surface diverges. These correlations are studied in the framework of Landau theory. It is shown the t^–1/2 at all types of transitions for systems with short-range forces wheret measures the distance from bulk coexistence.     

Special conformal symmetries in general relativity

A geometrical discussion of special conformal vector fields in space-time is given. In particular, it is shown that if such a vector field is admitted, it is unique up to a constant scaling and the addition of a homothetic or a Killing vector field. In the case when the gradient of the conformal scalar associated with is non-null it is shown that other homothetic and affine symmetries are necessarily admitted by the space-time, that an intrinsic family of 2-dimensional flat submanifolds is determined in the space-time, that is, in general, hypersurface orthogonal and that the space-time, if non-flat, is necessarily (geodesically) incomplete. Other geometrical features of such space-times are also considered.     

Screening Effect Due to Heavy Lower Tails in One-Dimensional Parabolic Anderson Model

We consider the large-time behavior of the solution to the parabolic Anderson problem _tu=u+u with initial data u(0, ·)=1 and non-positive finite i.i.d. potentials . Unlike in dimensions d2, the almost-sure decay rate of u(t, 0) as t is not determined solely by the upper tails of (0); too heavy lower tails of (0) accelerate the decay. The interpretation is that sites x with large negative (x) hamper the mass flow and hence screen off the influence of more favorable regions of the potential. The phenomenon is unique to d=1. The result answers an open question from our previous study [BK00] of this model in general dimension.     

Attractor States and Quantum Instabilities in de Sitter Space

The asymptotic behavior of the energy–momentum tensor for a free quantized scalar field with mass m and curvature coupling in de Sitter space is investigated. It is shown that for an arbitrary, homogeneous, and isotropic, fourth-order adiabatic state for which the two-point function is infrared finite, T _ab approaches the Bunch–Davies de Sitter invariant value at late times if m ² + R > 0. In the case m = = 0, the energy–momentum tensor approaches the de Sitter invariant Allen–Folacci value for such a state. For m ² + R = 0 but m and not separately zero, it is shown that at late times T _ab grows linearly in terms of cosmic time leading to an instability of de Sitter space. The asymptotic behavior is again independent of the state of the field. For m ² + R < 0, it is shown that, for most values of m and , T _ab grows exponentially in terms of cosmic time at late times in a state dependent manner.     

Integration in the GHP Formalism IV: A New Lie Derivative Operator Leading to an Efficient Treatment of Killing Vectors

In order to achieve efficient calculations and easy interpretations of symmetries, a strategy for investigations in tetrad formalisms is outlined: work in an intrinsic tetrad using intrinsic coordinates. The key result is that a vector field is a Killing vector field if and only if there exists a tetrad which is Lie derived with respect to ; this result is translated into the GHP formalism using a new generalised Lie derivative operator with respect to a vector field . We identify a class of it intrinsic GHP tetrads, which belongs to the class of GHP tetrads which is generalised Lie derived by this new generalised Lie derivative operator in the presence of a Killing vector field . This new operator also has the important property that, with respect to an intrinsic GHP tetrad, it commutes with the usual GHP operators if and only if is a Killing vector field. Practically, this means, for any spacetime obtained by integration in the GHP formalism using an intrinsic GHP tetrad, that the Killing vector properties can be deduced from the tetrad or metric using the Lie-GHP commutator equations, without a detailed additional analysis. Killing vectors are found in this manner for a number of special spaces.     

Solvable isotherms for a two-component system of charged rods on a line

The Coulomb system consisting of an equal number of positive and negative charged rods confined to a one-dimensional lattice is studied. The grand partition function can be calculated exactly at two values of the coupling constant=q ²/k _B T (q denoting the magnitude of the charges). The exact results lead to the conjecture that in the complex scaled fugacity plane, all the zeros of the grand partition function lie on the negative real axis for<2, on the point=–1 for=2, and on the unit circle for>2. In addition, for>4, we conjecture in general and prove at=4 that the zeros pinch the real axis in the thermodynamic limit, with an essential singularity in the pressure at the reduced density 1/2.     

Functional Realization and Nonlinear Induced Representation in the Geometrodifferential Conception of Extended Particles

In a model of extended particles described by Minkowski space-time variables x, de Sitter internal variables , a physical wave _x () representing the proper characteristics of the particles, and a functional wave X [ ] giving previsions, we study functional propagation of X in the space of physical waves (as advocated by a quantum functional theory) as well as the nonlinear realization of the internal de Sitter group on its Lorentz subgroup (introduced by Drechsler). The first study is undertaken in a special instance _x () = (x), while in the second the general structure of the model is adopted and curved space-time treated, but the functional propagation is not considered. A fiber bundle structure and an induced representation method are used. Propagators are derived, a quantum version of a variant of Drechsler's theory is obtained, and a nonlinear version of our model is constructed.     

Rotating steady-state configurations in general relativity. II

Space-times with timelike Killing vector field and axial Killing vector field are studied. Physical coordinates are constructed for the metric of differentially rotating matter. It is proved that, for matter flow whose streamline tangents areu = + , the matter region must be either Petrov type I orD.Partially supported by a National Research Council of Canada grant.     

Kinetics of random sequential parking on a line

We study the kinetics of irreversible random sequential parking of intervals of different sizes on an infinite line. For the simplest fixed-length parking distribution the model reduces to the known car-parking problem and we present an alternate solution to this problem. We also consider the general homogeneous case when the parking distribution varies asx ^–1 atx 1 with the lengthx of the filling interval. We develop a scaling theory describing such mixture-deposition processes and show that the scaled hole-size distribution(), with =xt ^z a scaling variable, decays with the scaled mass as ^–exp(—const·¹⁺) as . We determine scaling exponentsz and, and find that at large times the coverage(t) has a power-law form 1 – (t)t ^–v with nonuniversal exponent =(2–)/(1+) depending on the homogeneity index .