Nontrivial Function Research Articles

Random trees between two walls: exact partition function

We derive the exact partition function for a discrete model of random trees embedded in a one-dimensional space. These trees have vertices labeled by integers representing their position in the target space, with the SOS constraint that adjacent vertices have labels differing by +1 or -1. A non-trivial partition function is obtained whenever the target space is bounded by walls. We concentrate on the two cases where the target space is (i) the half-line bounded by a wall at the origin or (ii) a segment bounded by two walls at a finite distance. The general solution has a soliton-like structure involving elliptic functions. We derive the corresponding continuum scaling limit which takes the remarkable form of the Weierstrass p-function with constrained periods. These results are used to analyze the probability for an evolving population spreading in one dimension to attain the boundary of a given domain with the geometry of the target (i) or (ii). They also translate, via suitable bijections, into generating functions for bounded planar graphs.

Solvable models of strings in homogeneous plane-wave backgrounds

We solve closed string theory in all regular homogeneous plane-wave backgrounds with homogeneous NS three-form field strength and a dilaton. The parameters of the model are constant symmetric and anti-symmetric matrices k ij and f ij associated with the metric, and a constant anti-symmetric matrix h ij associated with the NS field strength. In the light-cone gauge the rotation parameters f ij have a natural interpretation as a constant magnetic field. This is a generalisation of the standard Landau problem with oscillator energies now being non-trivial functions of the parameters f ij and k ij . We develop a general procedure for solving linear but non-diagonal equations for string coordinates, and determine the corresponding oscillator frequencies, the light-cone Hamiltonian and level matching condition. We investigate the resulting string spectrum in detail in the four-dimensional case and compare the results with previously studied examples. Throughout we will find that the presence of the rotation parameter f ij can lead to certain unusual and unexpected features of the string spectrum like new massless states at non-zero string levels, stabilisation of otherwise unstable (tachyonic) modes, and discrete but not positive definite string oscillator spectra.

Monte Carlo sampling of negative-temperature plasma states

A Monte Carlo procedure is used to generate N-particle configurations compatible with two-temperature canonical equilibria in two dimensions, with particular attention to nonlinear plasma gyrokinetics. An unusual feature of the problem is the importance of a nontrivial probability density function P0(phi), the probability of realizing a set phi of Fourier amplitudes associated with an ensemble of uniformly distributed, independent particles. This quantity arises because the equilibrium distribution is specified in terms of phi, whereas the sampling procedure naturally produces particle states Gamma; phi and Gamma are related via a gyrokinetic Poisson equation, highly nonlinear in its dependence on Gamma. Expansion and asymptotic methods are used to calculate P0(phi) analytically; excellent agreement is found between the large-N asymptotic result and a direct numerical calculation. The algorithm is tested by successfully generating a variety of states of both positive and negative temperature, including ones in which either the longest- or shortest-wavelength modes are excited to relatively large amplitudes.

On the number of solutions for the two-point boundary value problem on Riemannian manifolds

We study the solutions joining two fixed points of a time-independent dynamical system on a Riemannian manifold ( M, g) from an enumerative point of view. We prove a finiteness result for solutions joining two points p, q∈ M that are non-conjugate in a suitable sense, under the assumption that ( M, g) admits a non-trivial convex function. We discuss in some detail the notion of conjugacy induced by a general dynamical system on a Riemannian manifold. Using techniques of infinite dimensional Morse theory on Hilbert manifolds we also prove that, under generic circumstances, the number of solutions joining two fixed points is odd. We present some examples where our theory applies.

Semianalytical solution of the Kondo model in a magnetic field

The single impurity Kondo model at zero temperature in a magnetic field is solved by a semi-analytical approach based on the flow equation method. The resulting problem is shown to be equivalent to a resonant level model with a non-constant hybridization function. This nontrivial effective hybridization function encodes the quasiparticle interaction in the Kondo limit, while the magnetic field enters as the impurity orbital energy. The evaluation of static and dynamic quantities of the strong-coupling Kondo model becomes very simple in this effective model. We present results for thermodynamic quantities and the dynamical spin-structure factor and compare them with NRG calculations.

Fractional quantum Hall effect, Jain’s rule, and topological textures

It is demonstrated that Jain’s rule for determining fractions in the quantum Hall effect can be obtained without recourse to the phenomenological concept of composite fermions. The possibility of existence is considered for topologically nontrivial many-electron wave functions, whose group classification gives an indication of special values of electron density in the ground states separated by a gap from excited energies.

Nonlinear effects induced by vacuum polarization by a strong electromagnetic field in a weak static electromagnetic field

In the one-loop approximation of quantum electrodynamics, we study the Faraday rotation effect in the propagation of a weak linearly polarized wave through the field of a strong plane electromagnetic wave and also processes induced by a strong wave: the emission of a photon whose energy is an integer multiple of the “strong-wave quantum” and photon absorption with the emission of several photons identical (coherent) to the “strong-wave quanta.” The induced processes should occur in a vacuum in a superposition of external fields, namely, the field of a strong plane electromagnetic wave and a static spatially inhomogeneous electromagnetic field. We show that the induced photon emission in the field of a strong electromagnetic wave with circular polarization and in a static inhomogeneous field is equivalent to the merging of two “strong-wave quanta” with the production of one photon and that the induced photon absorption in the same combination of fields is equivalent to the decay (splitting) of a photon into two photons completely identical to the “strong-wave quanta.” All these effects are induced by the phenomenon of the vacuum polarization by a strong external time-dependent electromagnetic field. The probabilities of the induced processes are nontrivial nonlinear functions of the squared strength of the time-dependent field also depending on other parameters.

Semilinear Perturbations of Harmonic Spaces, Liouville Property and a Boundary Value Problem

Let (X,ℋ) be a P-harmonic Bauer space and let Ψ be a Borel measurable function on X×R satisfying conditions (A) through (D) of Section 2 (e.g., Ψ(x,t)=t|t|α−1 where α>1). For every Kato family M of potential kernels on X let M U(X) denote the set of all real continuous functions on X such that u+K M D Ψ(⋅,u)∈ℋ(D) for every open relatively compact subset D of X. We study the existence of a non-trivial function in M U(X) which is dominated by a given positive harmonic function on X. If X is a domain of R d , μ is a positive Kato measure on X and L is a second-order differential operator in R d , we apply our study to derive a characterization of finite positive measures ν on the minimal Martin boundary ∂ M 1X for which the boundary value problem Lu=Ψ(⋅,u)μ in X and u=ν on ∂ M 1X is solvable.

Chiral Phase Transitions in QED at Finite Temperature: Dyson-Schwinger Equation Analysis in the Real Time Hard-Thermal-Loop Approximation

In order for clarifying what are the essential thermal effects that govern the chiral phase transition at finite temperature, we investigate, in the real-time thermal QED, the consequences of the Hard-Thermal-Loop (HTL) resummed Dyson-Schwinger equation for the physical fermion mass function $\Sigma_R$. Since $\Sigma_R$ is the mass function of an ``unstable'' quasi-particle in thermal field theories, it necessarily has non-trivial imaginary parts together with non-trivial wave function renormalization constants. In the present analysis we correctly respect this fact, and study, in the ladder approximation, the effect of HTL resummed gauge boson propagator. Our results with the use of numerical analysis, show the two facts; i) The chiral phase transition is of second order, since the fermion mass is dynamically generated at a critical value of the temperature $T_c$, or at the critical coupling constant $\alpha_c$, without any discontinuity, and ii) the critical temperature $T_c$ at fixed value of $\alpha$ is significantly lower than the previous results, namely the restoration of chiral symmetry occurs at lower temperature than previously expected. The second fact shows the importance of correctly taking the essential thermal effect into the analysis of chiral phase transition, which are, in the previous analyses, neglected due to the inappropriate approximations. The procedure how to maximally respect the gauge invariance in the present approximation, is also discussed.

The local time distribution of a particle diffusing on a graph

We study the local time distribution of a Brownian particle diffusing along the links on a graph. In particular, we derive an analytic expression of its Laplace transform in terms of the Green's function on the graph. We show that the asymptotic behavior of this distribution has non-Gaussian tails characterized by a nontrivial large deviation function.

On Correspondence of BRST-BFV, Dirac, and Refined Algebraic Quantizations of Constrained Systems

The correspondence between BRST-BFV, Dirac, and refined algebraic (group averaging, projection operator) approaches to quantizing constrained systems is analyzed. For the closed-algebra case, it is shown that the component of the BFV wave function corresponding to maximal (minimal) value of number of ghosts and antighosts in the Schrodinger representation may be viewed as a wave function in the refined algebraic (Dirac) quantization approach. The Giulini–Marolf group averaging formula for the inner product in the refined algebraic quantization approach is obtained from the Batalin–Marnelius prescription for the BRST-BFV inner product, which should be generally modified due to topological problems. The considered prescription for the correspondence of states is observed to be applicable to the open-algebra case. The refined algebraic quantization approach is generalized then to the case of nontrivial structure functions. A simple example is discussed. The correspondence of observables for different quantization methods is also investigated.

Prandtl-number dependence of interior temperature and velocity fluctuations in turbulent convection.

Temperature and vertical velocity fluctuations are measured in turbulent Rayleigh-Bénard convection at the center of an approximately unit aspect ratio container of cylindrical cross section. Our measurements show that the Rayleigh-number scaling exponent gamma of the interior temperature fluctuations (i.e., sigma(T)/deltaT approximately Ragamma) is a strong and nontrivial function of the Prandtl number in the range 2.9<Pr<12.4. Other measurements at constant Ra=2.0 x 10(9) show that the interior turbulent fluctuations decrease significantly with increasing Pr; the temperature and velocity fluctuations decrease by about approximately 45% and approximately 68% over our range in Pr.

Exact occupation time distribution in a non-Markovian sequence and its relation to spin glass models.

We compute exactly the distribution of the occupation time in a discrete non-Markovian toy sequence that appears in various physical contexts such as the diffusion processes and Ising spin glass chains. The non-Markovian property makes the results nontrivial even for this toy sequence. The distribution is shown to have non-Gaussian tails characterized by a nontrivial large deviation function that is computed explicitly. An exact mapping of this sequence to an Ising spin glass chain via a gauge transformation raises an interesting question for a generic finite sized spin glass model; at a given temperature, what is the distribution (over disorder) of the thermally averaged number of spins that are aligned to their local fields? We show that this distribution remains nontrivial even at infinite temperature and can be computed explicitly in few cases such as in the Sherrington-Kirkpatrick model with Gaussian disorder.

Theoretical Spectral Models of T Dwarfs at Short Wavelengths and Their Comparison with Data

We have generated new, self-consistent spectral and atmosphere models for the effective temperature range 600-1300 K thought to encompass the known T dwarfs. For the first time, theoretical models are compared with a family of measured T dwarf spectra at wavelengths shortward of ~1.0 μm. By defining spectral indices and standard colors in the optical and very near-infrared, we explore the theoretical systematics with Teff, gravity, and metallicity. We conclude that the short-wavelength range is rich in diagnostics that complement those in the near-infrared now used for spectral subtyping. We also conclude that the wings of the Na D and K I (7700 A) resonance lines and aggressive rainout of heavy metals (with the resulting enhancement of the sodium and potassium abundances at altitude) are required to fit the new data shortward of 1.0 μm. Furthermore, we find that the water bands weaken with increasing gravity, that modest decreases in metallicity enhance the effect in the optical of the sodium and potassium lines, and that at low values of Teff, in a reversal of the normal pattern, optical spectra become bluer with further decreases in Teff. Moreover, we conclude that T dwarf subtype is not a function of Teff alone but that it is a nontrivial function of gravity and metallicity as well. As do Marley and coworkers in their 2002 work, we see evidence in early T dwarf atmospheres of a residual effect of clouds. With cloudless models, we obtain spectral fits to the two late T dwarfs with known parallaxes, but a residual effect of clouds on the emergent spectra of even late T dwarfs cannot yet be discounted. However, our focus is not on detailed fits to individual objects but on the interpretation of the overall spectral and color trends of the entire class of T dwarfs, as seen at shorter wavelengths.

Response of non-equilibrium systems with long-range initial correlations

The long-time dynamics of the d-dimensional spherical model with a non-conserved order parameter and quenched from an initial state with long-range correlations is studied through the exact calculation of the two-time autocorrelation and autoresponse functions. In the ageing regime, these are given in terms of non-trivial universal scaling functions of both time variables. At criticality, five distinct types of ageing are found, depending on the form of the initial correlations, while at low temperatures only a single type of ageing exists. The autocorrelation and autoresponse exponents are shown to be generically different and to depend on the initial conditions. The scaling form of the two-time response functions agrees with a recent prediction coming from conformal invariance.

Field-induced spin-sensity-wave phases in quasi-one-dimensional conductors: theory versus experiments.

We show that the "quantized nesting" model misses important features of the magnetic-field-induced spin-density-wave (FISDW) phase diagram. Among them are the following: (1) the FISDW wave vector is not strictly quantized; (2) in some compounds, the FISDW diagram consists of two regions: (a) At low temperatures, there are jumps of the wave vector (i.e., the first order transitions between FISDW phases); (b) at high temperatures the jumps and the first order transitions disappear, but the wave vector is still a nontrivial function of a magnetic field. These are in agreement with the experiments on (TMTSF)2PF6.

The Validity of the Multifractal Formalism: Results and Examples

By obtaining a new sufficient condition for a valid multifractal formalism, we improve in this paper a result developed by L. Olsen (1995, Adv. Math.116, 82–196). In particular, we describe a large class of measures satisfying the multifractal formalism and for which the construction of Gibbs measures is not possible. Some of these measures are not unidimensional but have a nontrivial multifractal spectrum, giving a negative answer to a question asked by S. J. Taylor (1995, J. Fourier Anal. Appl., special issue). We also describe a necessary condition of validity for the formalism which is very close to the sufficient one. This necessary condition allows us to describe a measure μ for which the multifractal packing dimension function Bμ(q) is a nontrivial real analytic function but the multifractal formalism is nowhere satisfied. This example gives also a solution to a problem posed by Taylor (cited above).

Total Scalar Curvature and L 2 Harmonic 1-Forms on a Minimal Hypersurface in Euclidean Space

Let M n , n ≥ 3, be a complete oriented immersed minimal hypersurface in Euclidean space R n+1. We show that if the total scalar curvature on M is less than the n/2 power of 1/C s , where C s is the Sobolev constant for M, then there are no L 2 harmonic 1-forms on M. As corollaries, such a minimal hypersurface contains no nontrivial harmonic functions with finite Dirichlet integral and so it has only one end. This implies finally that M is a hyperplane.

Structure of radiatively induced Lorentz andCPTviolation in QED at finite temperature

We obtain the induced Lorentz- and CPT-violating term in QED at finite temperature using imaginary-time formalism and dimensional regularization. Its form resembles a Chern-Simons-like structure, but, unexpectedly, it does not depend on the temporal component of the fixed $b_\mu$ constant vector that is coupled to the axial current. Nevertheless Ward identities are respected and its coefficient vanishes at T=0, consistently with previous computations with the same regularization procedure, and it is a non-trivial function of temperature. We argue that at finite $T$ a Chern-Simons-like Lorentz- and CPT-violating term is generically present, the value of its coefficient being unambiguously determined up to a $T-$independent constant, related to the zero-temperature renormalization conditions.

More exact predictions of SUSYM for string theory

We compute the coefficients of an infinite family of chiral primary operators in the local operator expansion of a circular Wilson loop in N=4 supersymmetric Yang–Mills theory. The computation sums all planar rainbow Feynman graphs. We argue that radiative corrections from planar graphs with internal vertices cancel in leading orders and we conjecture that they cancel to all orders in perturbation theory. The coefficients are nontrivial functions of the 't Hooft coupling and their strong coupling limits are in exact agreement with those previously computed using the AdS/CFT correspondence. They predict the subleading orders in strong coupling and could in principle be compared with string theory calculations.