Field theory of directed percolation with long-range spreading

It is well established that the phase transition between survival and extinction in spreading models with short-range interactions is generically associated with the directed percolation (DP) universality class. In many realistic spreading processes, however, interactions are long ranged and well described by L\'{e}vy-flights, i.e., by a probability distribution that decays in $d$ dimensions with distance $r$ as $r^{-d-\sigma}$. We employ the powerful methods of renormalized field theory to study DP with such long range, L\'{e}vy-flight spreading in some depth. Our results unambiguously corroborate earlier findings that there are four renormalization group fixed points corresponding to, respectively, short-range Gaussian, L\'{e}vy Gaussian, short-range DP and L\'{e}vy DP, and that there are four lines in the $(\sigma, d)$ plane which separate the stability regions of these fixed points. When the stability line between short-range DP and L\'{e}vy DP is crossed, all critical exponents change continuously. We calculate the exponents describing L\'{e}vy DP to second order in $\epsilon$-expansion, and we compare our analytical results to the results of existing numerical simulations. Furthermore, we calculate the leading logarithmic corrections for several dynamical observables.Comment: 12 pages, 3 figure

Finite-size scaling of directed percolation above the upper critical dimension

We consider analytically as well as numerically the finite-size scaling behavior in the stationary state near the non-equilibrium phase transition of directed percolation within the mean field regime, i.e., above the upper critical dimension. Analogous to equilibrium, usual finite-size scaling is valid below the upper critical dimension, whereas it fails above. Performing a momentum analysis of associated path integrals we derive modified finite-size scaling forms of the order parameter and its higher moments. The results are confirmed by numerical simulations of corresponding high-dimensional lattice models.Comment: 4 pages, one figur

A Generalized Epidemic Process and Tricritical Dynamic Percolation

The renowned general epidemic process describes the stochastic evolution of a population of individuals which are either susceptible, infected or dead. A second order phase transition belonging to the universality class of dynamic isotropic percolation lies between endemic or pandemic behavior of the process. We generalize the general epidemic process by introducing a fourth kind of individuals, viz. individuals which are weakened by the process but not yet infected. This sensibilization gives rise to a mechanism that introduces a global instability in the spreading of the process and therefore opens the possibility of a discontinuous transition in addition to the usual continuous percolation transition. The tricritical point separating the lines of first and second order transitions constitutes a new universality class, namely the universality class of tricritical dynamic isotropic percolation. Using renormalized field theory we work out a detailed scaling description of this universality class. We calculate the scaling exponents in an $\epsilon$-expansion below the upper critical dimension $d_{c}=5$ for various observables describing tricritical percolation clusters and their spreading properties. In a remarkable contrast to the usual percolation transition, the exponents $\beta$ and ${\beta}^{\prime}$ governing the two order parameters, viz. the mean density and the percolation probability, turn out to be different at the tricritical point. In addition to the scaling exponents we calculate for all our static and dynamic observables logarithmic corrections to the mean-field scaling behavior at $d_c=5$.Comment: 21 pages, 10 figures, version to appear in Phys. Rev.

Finite-size scaling of directed percolation in the steady state

Recently, considerable progress has been made in understanding finite-size scaling in equilibrium systems. Here, we study finite-size scaling in non-equilibrium systems at the instance of directed percolation (DP), which has become the paradigm of non-equilibrium phase transitions into absorbing states, above, at and below the upper critical dimension. We investigate the finite-size scaling behavior of DP analytically and numerically by considering its steady state generated by a homogeneous constant external source on a d-dimensional hypercube of finite edge length L with periodic boundary conditions near the bulk critical point. In particular, we study the order parameter and its higher moments using renormalized field theory. We derive finite-size scaling forms of the moments in a one-loop calculation. Moreover, we introduce and calculate a ratio of the order parameter moments that plays a similar role in the analysis of finite size scaling in absorbing nonequilibrium processes as the famous Binder cumulant in equilibrium systems and that, in particular, provides a new signature of the DP universality class. To complement our analytical work, we perform Monte Carlo simulations which confirm our analytical results.Comment: 21 pages, 6 figure

Spontaneous Symmetry Breaking in Directed Percolation with Many Colors: Differentiation of Species in the Gribov Process

A general field theoretic model of directed percolation with many colors that is equivalent to a population model (Gribov process) with many species near their extinction thresholds is presented. It is shown that the multicritical behavior is always described by the well known exponents of Reggeon field theory. In addition this universal model shows an instability that leads in general to a total asymmetry between each pair of species of a cooperative society.Comment: 4 pages, 2 Postscript figures, uses multicol.sty, submitte

Short-time Critical Dynamics of the 3-Dimensional Ising Model

Comprehensive Monte Carlo simulations of the short-time dynamic behaviour are reported for the three-dimensional Ising model at criticality. Besides the exponent $\theta$ of the critical initial increase and the dynamic exponent $z$, the static critical exponents $\nu$ and $\beta$ as well as the critical temperature are determined from the power-law scaling behaviour of observables at the beginning of the time evolution. States of very high temperature as well as of zero temperature are used as initial states for the simulations.Comment: 8 pages with 7 figure

Local scale invariance in the parity conserving nonequilibrium kinetic Ising model

The local scale invariance has been investigated in the nonequilibrium kinetic Ising model exhibiting absorbing phase transition of PC type in 1+1 dimension. Numerical evidence has been found for the satisfaction of this symmetry and estimates for the critical ageing exponents are given.Comment: 8 pages, 2 figures (IOP format), final form to appear in JSTA

Forces on Bins - The Effect of Random Friction

In this note we re-examine the classic Janssen theory for stresses in bins, including a randomness in the friction coefficient. The Janssen analysis relies on assumptions not met in practice; for this reason, we numerically solve the PDEs expressing balance of momentum in a bin, again including randomness in friction.Comment: 11 pages, LaTeX, with 9 figures encoded, gzippe

Quantized charge pumping through a quantum dot by surface acoustic waves

We present a realization of quantized charge pumping. A lateral quantum dot is defined by metallic split gates in a GaAs/AlGaAs heterostructure. A surface acoustic wave whose wavelength is twice the dot length is used to pump single electrons through the dot at a frequency f=3GHz. The pumped current shows a regular pattern of quantization at values I=nef over a range of gate voltage and wave amplitude settings. The observed values of n, the number of electrons transported per wave cycle, are determined by the number of electronic states in the quantum dot brought into resonance with the fermi level of the electron reservoirs during the pumping cycle.Comment: 8 page

Localization of Multi-Dimensional Wigner Distributions

A well known result of P. Flandrin states that a Gaussian uniquely maximizes the integral of the Wigner distribution over every centered disc in the phase plane. While there is no difficulty in generalizing this result to higher-dimensional poly-discs, the generalization to balls is less obvious. In this note we provide such a generalization.Comment: Minor corrections, to appear in the Journal of Mathematical Physic