Scaling of loop-erased walks in 2 to 4 dimensions

We simulate loop-erased random walks on simple (hyper-)cubic lattices of dimensions 2,3, and 4. These simulations were mainly motivated to test recent two loop renormalization group predictions for logarithmic corrections in $d=4$, simulations in lower dimensions were done for completeness and in order to test the algorithm. In $d=2$, we verify with high precision the prediction $D=5/4$, where the number of steps $n$ after erasure scales with the number $N$ of steps before erasure as $n\sim N^{D/2}$. In $d=3$ we again find a power law, but with an exponent different from the one found in the most precise previous simulations: $D = 1.6236\pm 0.0004$. Finally, we see clear deviations from the naive scaling $n\sim N$ in $d=4$. While they agree only qualitatively with the leading logarithmic corrections predicted by several authors, their agreement with the two-loop prediction is nearly perfect.Comment: 3 pages, including 3 figure

Time-Fractional KdV Equation: Formulation and Solution using Variational Methods

In this work, the semi-inverse method has been used to derive the Lagrangian of the Korteweg-de Vries (KdV) equation. Then, the time operator of the Lagrangian of the KdV equation has been transformed into fractional domain in terms of the left-Riemann-Liouville fractional differential operator. The variational of the functional of this Lagrangian leads neatly to Euler-Lagrange equation. Via Agrawal's method, one can easily derive the time-fractional KdV equation from this Euler-Lagrange equation. Remarkably, the time-fractional term in the resulting KdV equation is obtained in Riesz fractional derivative in a direct manner. As a second step, the derived time-fractional KdV equation is solved using He's variational-iteration method. The calculations are carried out using initial condition depends on the nonlinear and dispersion coefficients of the KdV equation. We remark that more pronounced effects and deeper insight into the formation and properties of the resulting solitary wave by additionally considering the fractional order derivative beside the nonlinearity and dispersion terms.Comment: The paper has been rewritten, 12 pages, 3 figure

Field theory conjecture for loop-erased random walks

We give evidence that the functional renormalization group (FRG), developed to study disordered systems, may provide a field theoretic description for the loop-erased random walk (LERW), allowing to compute its fractal dimension in a systematic expansion in epsilon=4-d. Up to two loop, the FRG agrees with rigorous bounds, correctly reproduces the leading logarithmic corrections at the upper critical dimension d=4, and compares well with numerical studies. We obtain the universal subleading logarithmic correction in d=4, which can be used as a further test of the conjecture.Comment: 5 page

Constant Curvature Coefficients and Exact Solutions in Fractional Gravity and Geometric Mechanics

We study fractional configurations in gravity theories and Lagrange mechanics. The approach is based on Caputo fractional derivative which gives zero for actions on constants. We elaborate fractional geometric models of physical interactions and we formulate a method of nonholonomic deformations to other types of fractional derivatives. The main result of this paper consists in a proof that for corresponding classes of nonholonomic distributions a large class of physical theories are modelled as nonholonomic manifolds with constant matrix curvature. This allows us to encode the fractional dynamics of interactions and constraints into the geometry of curve flows and solitonic hierarchies.Comment: latex2e, 11pt, 27 pages, the variant accepted to CEJP; added and up-dated reference

Conservative Constraints on Dark Matter from the Fermi-LAT Isotropic Diffuse Gamma-Ray Background Spectrum

We examine the constraints on final state radiation from Weakly Interacting Massive Particle (WIMP) dark matter candidates annihilating into various standard model final states, as imposed by the measurement of the isotropic diffuse gamma-ray background by the Large Area Telescope aboard the Fermi Gamma-Ray Space Telescope. The expected isotropic diffuse signal from dark matter annihilation has contributions from the local Milky Way (MW) as well as from extragalactic dark matter. The signal from the MW is very insensitive to the adopted dark matter profile of the halos, and dominates the signal from extragalactic halos, which is sensitive to the low mass cut-off of the halo mass function. We adopt a conservative model for both the low halo mass survival cut-off and the substructure boost factor of the Galactic and extragalactic components, and only consider the primary final state radiation. This provides robust constraints which reach the thermal production cross-section for low mass WIMPs annihilating into hadronic modes. We also reanalyze limits from HESS observations of the Galactic Ridge region using a conservative model for the dark matter halo profile. When combined with the HESS constraint, the isotropic diffuse spectrum rules out all interpretations of the PAMELA positron excess based on dark matter annihilation into two lepton final states. Annihilation into four leptons through new intermediate states, although constrained by the data, is not excluded.Comment: 11 pages, 5 figures. v3: minor revisions, matches version to appear in JCA

Red Queen Coevolution on Fitness Landscapes

Species do not merely evolve, they also coevolve with other organisms. Coevolution is a major force driving interacting species to continuously evolve ex- ploring their fitness landscapes. Coevolution involves the coupling of species fit- ness landscapes, linking species genetic changes with their inter-specific ecological interactions. Here we first introduce the Red Queen hypothesis of evolution com- menting on some theoretical aspects and empirical evidences. As an introduction to the fitness landscape concept, we review key issues on evolution on simple and rugged fitness landscapes. Then we present key modeling examples of coevolution on different fitness landscapes at different scales, from RNA viruses to complex ecosystems and macroevolution.Comment: 40 pages, 12 figures. To appear in "Recent Advances in the Theory and Application of Fitness Landscapes" (H. Richter and A. Engelbrecht, eds.). Springer Series in Emergence, Complexity, and Computation, 201

Magnetism in Dense Quark Matter

We review the mechanisms via which an external magnetic field can affect the ground state of cold and dense quark matter. In the absence of a magnetic field, at asymptotically high densities, cold quark matter is in the Color-Flavor-Locked (CFL) phase of color superconductivity characterized by three scales: the superconducting gap, the gluon Meissner mass, and the baryonic chemical potential. When an applied magnetic field becomes comparable with each of these scales, new phases and/or condensates may emerge. They include the magnetic CFL (MCFL) phase that becomes relevant for fields of the order of the gap scale; the paramagnetic CFL, important when the field is of the order of the Meissner mass, and a spin-one condensate associated to the magnetic moment of the Cooper pairs, significant at fields of the order of the chemical potential. We discuss the equation of state (EoS) of MCFL matter for a large range of field values and consider possible applications of the magnetic effects on dense quark matter to the astrophysics of compact stars.Comment: To appear in Lect. Notes Phys. "Strongly interacting matter in magnetic fields" (Springer), edited by D. Kharzeev, K. Landsteiner, A. Schmitt, H.-U. Ye

Reduction of the value of information sharing as demand becomes strongly auto-correlated

Information sharing has been identified, in the academic literature, as one of the most important levers to mitigate the bullwhip effect in supply chains. A highly-cited article on the bullwhip effect has claimed that the percentage inventory reduction resulting from information sharing in a two level supply chain, when the downstream demand is autoregressive of order one, is an increasing function of the autoregressive parameter of the demand. In this paper we show that this is true only for a certain range of the autoregressive parameter and there is a maximum value beyond which the bullwhip ratio at the upstream stage is reduced and the percentage inventory reduction resulting from information sharing decreases towards zero. We also show that this maximum value of the autoregressive parameter can be as high as 0.7 which represents a common value that may be encountered in many practical contexts. This means that large benefits of information sharing cannot be assumed for those Stock Keeping Units (SKUs) with highly positively auto-correlated demand. Instead, equally careful analysis is needed for these items as for those SKUs with less strongly auto-correlated demand

Fractional Dynamics of Relativistic Particle

Fractional dynamics of relativistic particle is discussed. Derivatives of fractional orders with respect to proper time describe long-term memory effects that correspond to intrinsic dissipative processes. Relativistic particle subjected to a non-potential four-force is considered as a nonholonomic system. The nonholonomic constraint in four-dimensional space-time represents the relativistic invariance by the equation for four-velocity u_{\mu} u^{\mu}+c^2=0, where c is a speed of light in vacuum. In the general case, the fractional dynamics of relativistic particle is described as non-Hamiltonian and dissipative. Conditions for fractional relativistic particle to be a Hamiltonian system are considered