Hyperscaling in the Domany-Kinzel Cellular Automaton

An apparent violation of hyperscaling at the endpoint of the critical line in the Domany-Kinzel stochastic cellular automaton finds an elementary resolution upon noting that the order parameter is discontinuous at this point. We derive a hyperscaling relation for such transitions and discuss applications to related examples.Comment: 8 pages, latex, no figure

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

Cluster Approximation for the Contact Process

The one-dimensional contact process is analyzed by a cluster approximation. In this approach, the hierarchy of rate equations for the densities of finite length empty intervals are truncated under the assumption that adjacent intervals are not correlated. This assumption yields a first order phase transition from an active state to the adsorbing state. Despite the apparent failure of this approximation in describing the critical behavior, our approach provides an accurate description of the steady state properties for a significant range of desorption rates. Moreover, the resulting critical exponents are closer to the simulation values in comparison with site mean-field theory.Comment: 9 pages, Latex format, 2 postscript figure

Analysing and controlling the tax evasion dynamics via majority-vote model

Within the context of agent-based Monte-Carlo simulations, we study the well-known majority-vote model (MVM) with noise applied to tax evasion on simple square lattices, Voronoi-Delaunay random lattices, Barabasi-Albert networks, and Erd\"os-R\'enyi random graphs. In the order to analyse and to control the fluctuations for tax evasion in the economics model proposed by Zaklan, MVM is applied in the neighborhod of the noise critical $q_{c}$. The Zaklan model had been studied recently using the equilibrium Ising model. Here we show that the Zaklan model is robust and can be reproduced also through the nonequilibrium MVM on various topologies.Comment: 18 pages, 7 figures, LAWNP'09, 200

Generalized Scaling for Models with Multiple Absorbing States

At a continuous transition into a nonunique absorbing state, particle systems may exhibit nonuniversal critical behavior, in apparent violation of hyperscaling. We propose a generalized scaling theory for dynamic critical behavior at a transition into an absorbing state, which is capable of describing exponents which vary according to the initial configuration. The resulting hyperscaling relation is supported by simulations of two lattice models.Comment: Latex 9 page

Locating the minimum : Approach to equilibrium in a disordered, symmetric zero range process

We consider the dynamics of the disordered, one-dimensional, symmetric zero range process in which a particle from an occupied site $k$ hops to its nearest neighbour with a quenched rate $w(k)$. These rates are chosen randomly from the probability distribution $f(w) \sim (w-c)^{n}$, where $c$ is the lower cutoff. For $n > 0$, this model is known to exhibit a phase transition in the steady state from a low density phase with a finite number of particles at each site to a high density aggregate phase in which the site with the lowest hopping rate supports an infinite number of particles. In the latter case, it is interesting to ask how the system locates the site with globally minimum rate. We use an argument based on local equilibrium, supported by Monte Carlo simulations, to describe the approach to the steady state. We find that at large enough time, the mass transport in the regions with a smooth density profile is described by a diffusion equation with site-dependent rates, while the isolated points where the mass distribution is singular act as the boundaries of these regions. Our argument implies that the relaxation time scales with the system size $L$ as $L^{z}$ with $z=2+1/(n+1)$ for $n > 1$ and suggests a different behaviour for $n < 1$.Comment: Revtex, 7 pages including 3 figures. Submitted to Pramana -- special issue on mesoscopic and disordered system

Fluctuations of the one-dimensional asymmetric exclusion process using random matrix techniques

The studies of fluctuations of the one-dimensional Kardar-Parisi-Zhang universality class using the techniques from random matrix theory are reviewed from the point of view of the asymmetric simple exclusion process. We explain the basics of random matrix techniques, the connections to the polynuclear growth models and a method using the Green's function.Comment: 41 pages, 10 figures, minor corrections, references adde

A sufficient criterion for integrability of stochastic many-body dynamics and quantum spin chains

We propose a dynamical matrix product ansatz describing the stochastic dynamics of two species of particles with excluded-volume interaction and the quantum mechanics of the associated quantum spin chains respectively. Analyzing consistency of the time-dependent algebra which is obtained from the action of the corresponding Markov generator, we obtain sufficient conditions on the hopping rates for identifing the integrable models. From the dynamical algebra we construct the quadratic algebra of Zamolodchikov type, associativity of which is a Yang Baxter equation. The Bethe ansatz equations for the spectra are obtained directly from the dynamical matrix product ansatz.Comment: 19 pages Late

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

Exact Results for Kinetics of Catalytic Reactions

The kinetics of an irreversible catalytic reaction on substrate of arbitrary dimension is examined. In the limit of infinitesimal reaction rate (reaction-controlled limit), we solve the dimer-dimer surface reaction model (or voter model) exactly in arbitrary dimension $D$. The density of reactive interfaces is found to exhibit a power law decay for $D<2$ and a slow logarithmic decay in two dimensions. We discuss the relevance of these results for the monomer-monomer surface reaction model.Comment: 4 pages, RevTeX, no figure