Critical behavior for mixed site-bond directed percolation

We study mixed site-bond directed percolation on 2D and 3D lattices by using time-dependent simulations. Our results are compared with rigorous bounds recently obtained by Liggett and by Katori and Tsukahara. The critical fractions $p_{site}^c$ and $p_{bond}^c$ of sites and bonds are extremely well approximated by a relationship reported earlier for isotropic percolation, $(\log p_{site}^c/\log p_{site}^{c^*}+\log p_{bond}^c/\log p_{bond}^{c^*} = 1)$, where $p_{site}^{c^*}$ and $p_{bond}^{c^*}$ are the critical fractions in pure site and bond directed percolation.Comment: 10 pages, figures available on request from [email protected]

Numerical Diagonalisation Study of the Trimer Deposition-Evaporation Model in One Dimension

We study the model of deposition-evaporation of trimers on a line recently introduced by Barma, Grynberg and Stinchcombe. The stochastic matrix of the model can be written in the form of the Hamiltonian of a quantum spin-1/2 chain with three-spin couplings given by H= \sum\displaylimits_i [(1 - \sigma_i^-\sigma_{i+1}^-\sigma_{i+2}^-) \sigma_i^+\sigma_{i+1}^+\sigma_{i+2}^+ + h.c]. We study by exact numerical diagonalization of $H$ the variation of the gap in the eigenvalue spectrum with the system size for rings of size up to 30. For the sector corresponding to the initial condition in which all sites are empty, we find that the gap vanishes as $L^{-z}$ where the gap exponent $z$ is approximately $2.55\pm 0.15$. This model is equivalent to an interfacial roughening model where the dynamical variables at each site are matrices. From our estimate for the gap exponent we conclude that the model belongs to a new universality class, distinct from that studied by Kardar, Parisi and Zhang.Comment: 11 pages, 2 figures (included

Active Width at a Slanted Active Boundary in Directed Percolation

The width W of the active region around an active moving wall in a directed percolation process diverges at the percolation threshold p_c as W \simeq A \epsilon^{-\nu_\parallel} \ln(\epsilon_0/\epsilon), with \epsilon=p_c-p, \epsilon_0 a constant, and \nu_\parallel=1.734 the critical exponent of the characteristic time needed to reach the stationary state \xi_\parallel \sim \epsilon^{-\nu_\parallel}. The logarithmic factor arises from screening of statistically independent needle shaped sub clusters in the active region. Numerical data confirm this scaling behaviour.Comment: 5 pages, 5 figure

Directed Fixed Energy Sandpile Model

We numerically study the directed version of the fixed energy sandpile. On a closed square lattice, the dynamical evolution of a fixed density of sand grains is studied. The activity of the system shows a continuous phase transition around a critical density. While the deterministic version has the set of nontrivial exponents, the stochastic model is characterized by mean field like exponents.Comment: 5 pages, 6 figures, to be published in Phys. Rev.

Nature of phase transitions in a probabilistic cellular automaton with two absorbing states

We present a probabilistic cellular automaton with two absorbing states, which can be considered a natural extension of the Domany-Kinzel model. Despite its simplicity, it shows a very rich phase diagram, with two second-order and one first-order transition lines that meet at a tricritical point. We study the phase transitions and the critical behavior of the model using mean field approximations, direct numerical simulations and field theory. A closed form for the dynamics of the kinks between the two absorbing phases near the tricritical point is obtained, providing an exact correspondence between the presence of conserved quantities and the symmetry of absorbing states. The second-order critical curves and the kink critical dynamics are found to be in the directed percolation and parity conservation universality classes, respectively. The first order phase transition is put in evidence by examining the hysteresis cycle. We also study the "chaotic" phase, in which two replicas evolving with the same noise diverge, using mean field and numerical techniques. Finally, we show how the shape of the potential of the field-theoretic formulation of the problem can be obtained by direct numerical simulations.Comment: 19 pages with 7 figure

Nonequilibrium Dynamics and Aging in the Three--Dimensional Ising Spin Glass Model

The low temperature dynamics of the three dimensional Ising spin glass in zero field with a discrete bond distribution is investigated via MC simulations. The thermoremanent magnetization is found to decay algebraically and the temperature dependent exponents agree very well with the experimentally determined values. The nonequilibrium autocorrelation function $C(t,t_w)$ shows a crossover at the waiting (or {\em aging}) time $t_w$ from algebraic {\em quasi-equilibrium} decay for times $t$$\ll$$t_w$ to another, faster algebraic decay for $t$$\gg$$t_w$ with an exponent similar to one for the remanent magnetization.Comment: Revtex, 11 pages + 4 figures (included as Latex-files

A simple model of epitaxial growth

A discrete solid-on-solid model of epitaxial growth is introduced which, in a simple manner, takes into account the effect of an Ehrlich-Schwoebel barrier at step edges as well as the local relaxation of incoming particles. Furthermore a fast step edge diffusion is included in 2+1 dimensions. The model exhibits the formation of pyramid-like structures with a well-defined constant inclination angle. Two regimes can be distinguished clearly: in an initial phase (I) a definite slope is selected while the number of pyramids remains unchanged. Then a coarsening process (II) is observed which decreases the number of islands according to a power law in time. Simulations support self-affine scaling of the growing surface in both regimes. The roughness exponent is alpha =1 in all cases. For growth in 1+1 dimensions we obtain dynamic exponents z = 2 (I) and z = 3 (II). Simulations for d=2+1 seem to be consistent with z= 2 (I) and z= 2.3 (II) respectively.Comment: 8 pages Latex2e, 4 Postscript figures included, uses packages a4wide,epsfig,psfig,amsfonts,latexsy

Precise Critical Exponents for the Basic Contact Process

We calculated some of the critical exponents of the directed percolation universality class through exact numerical diagonalisations of the master operator of the one-dimensional basic contact process. Perusal of the power method together with finite-size scaling allowed us to achieve a high degree of accuracy in our estimates with relatively little computational effort. A simple reasoning leading to the appropriate choice of the microscopic time scale for time-dependent simulations of Markov chains within the so called quantum chain formulation is discussed. Our approach is applicable to any stochastic process with a finite number of absorbing states.Comment: LaTeX 2.09, 9 pages, 1 figur

A study of logarithmic corrections and universal amplitude ratios in the two-dimensional 4-state Potts model

Monte Carlo (MC) and series expansion (SE) data for the energy, specific heat, magnetization and susceptibility of the two-dimensional 4-state Potts model in the vicinity of the critical point are analysed. The role of logarithmic corrections is discussed and an approach is proposed in order to account numerically for these corrections in the determination of critical amplitudes. Accurate estimates of universal amplitude ratios $A_+/A_-$, $\Gamma_+/\Gamma_-$, $\Gamma_T/\Gamma_-$ and $R_C^\pm$ are given, which arouse new questions with respect to previous works

Unravelling quantum carpets: a travelling wave approach

Quantum carpets are generic spacetime patterns formed in the probability distributions P(x,t) of one-dimensional quantum particles, first discovered in 1995. For the case of an infinite square well potential, these patterns are shown to have a detailed quantitative explanation in terms of a travelling-wave decomposition of P(x,t). Each wave directly yields the time-averaged structure of P(x,t) along the (quantised)spacetime direction in which the wave propagates. The decomposition leads to new predictions of locations, widths depths and shapes of carpet structures, and results are also applicable to light diffracted by a periodic grating and to the quantum rotator. A simple connection between the waves and the Wigner function of the initial state of the particle is demonstrated, and some results for more general potentials are given.Comment: Latex, 26 pages + 6 figures, submitted to J. Phys. A (connections with prior literature clarified