Directed Percolation Universality Class Research Articles

Density matrix renormalization group and reaction-diffusion processes

The density matrix renormalization group ( DMRG) is applied to some one-dimensional reaction-diffusion models in the vicinity of and at their critical point. The stochastic time evolution for these models is given in terms of a non-symmetric “quantum Hamiltonian”, which is diagonalized using the DMRG method for open chains of moderate lengths (up to about 60 sites). The numerical diagonalization methods for non-symmetric matrices are reviewed. Different choices for an appropriate density matrix in the non-symmetric DMRG are discussed. Accurate estimates of the steady-state critical points and exponents can then be found from finite-size scaling through standard finite-lattice extrapolation methods. This is exemplified by studying the leading relaxation time and the density profiles of diffusion-annihilation and of a branching-fusing model in the directed percolation universality class.

Roughening transition of a restricted solid-on-solid model in the directed percolation universality class.

We carried out a finite-size scaling analysis of the restricted solid-on-solid version of a recently introduced growth model that exhibits a roughening transition accompanied by spontaneous symmetry breaking. The dynamic critical exponent of the model was calculated and found to be consistent with the universality class of the directed percolation process in a symmetry-broken phase with a crossover to Kardar-Parisi-Zhang behavior in a rough phase. The order parameter of the roughening transition together with the string order parameter was calculated, and we found that the flat, gapped phase is disordered with an antiferromagnetic spin-fluid structure of kinks, although strongly dominated by the completely flat configuration without kinks. A possible interesting extension of the model is mentioned.

Model of biological evolution with threshold dynamics and infinitely many absorbing states.

We study a model of biological evolution where the survival of a given species depends on its interactions with neighboring species. In the steady state the model has an active phase and an absorbing phase, which are separated by the critical point of the directed percolation universality class. The absorbing phase is infinitely degenerate and the dynamical behavior of our model is found to be nonuniversal.

Critical exponents for a dimer–dimer irreversible surface reaction

A dimer–dimer reaction model of the type A 2+B 2→2AB is investigated by means of computer simulations. The desorption of B 2 dimer with probability P is studied. Here we investigate the critical behaviour. The critical exponents of our model do not fall into the Reggeon field theory or directed percolation universality class. The universality class of the reaction changes with changes in P.

Interacting monomer-dimer model with infinitely many absorbing states

We study a modified version of the interacting monomer-dimer (IMD) model that has infinitely many absorbing (IMA) states. Unlike previously studied models with IMA states, the absorbing states can be divided into two equivalent groups which are dynamically separated infinitely far apart. Monte Carlo simulations show that this model belongs to a directed Ising universality class like the ordinary IMD model with two equivalent absorbing states. To our knowledge, this model is the first model with IMA states which does not belong to the directed percolation (DP) universality class. The DP universality class can be restored in two ways, i.e., by connecting the two equivalent groups dynamically, or by introducing a symmetry-breaking field between the two groups.

Synchronous versus asynchronous updating in the “game of Life”

The rules for the ‘‘game of Life’’ are modified to allow for only a random fraction of sites to be updated in each time step. Under variation of this fraction from the parallel updating limit down to the Poisson limit, a critical phase transition is observed that explains why the game of Life appears to obey self-organized criticality. The critical exponents are calculated and the static exponents appear to belong to the directed percolation universality class in 211 dimensions. The dynamic exponents, however, are nonuniversal, as seen in other systems with multiple absorbing states. @S1063-651X~99!13903-5#

Synchronization of coupled systems with spatiotemporal chaos

We argue that the synchronization transition of stochastically coupled cellular automata, discovered recently by L.G. Morelli {\it et al.} (Phys. Rev. {\bf 58 E}, R8 (1998)), is generically in the directed percolation universality class. In particular, this holds numerically for the specific example studied by these authors, in contrast to their claim. For real-valued systems with spatiotemporal chaos such as coupled map lattices, we claim that the synchronization transition is generically in the universality class of the Kardar-Parisi-Zhang equation with a nonlinear growth limiting term.

Moment ratios for absorbing-state phase transitions

We determine the first through fourth moments of the order parameter, and various ratios, for several one- and two-dimensional models with absorbing-state phase transitions. We perform a detailed analysis of the system-size dependence of these ratios, and confirm that they are indeed universal for three models - the contact process, the A model, and the pair contact process - belonging to the directed percolation universality class. Our studies also yield a refined estimate for the critical point of the pair contact process.

Critical behavior of a one-dimensional monomer-dimer reaction model with lateral interactions

A monomer-dimer reaction lattice model with lateral repulsion among the same species is studied using a mean-field analysis and Monte Carlo simulations. For weak repulsions, the model exhibits a first-order irreversible phase transition between two absorbing states saturated by each different species. Increasing the repulsion, a reactive stationary state appears in addition to the saturated states. The irreversible phase transitions from the reactive phase to any of the saturated states are continuous and belong to the directed percolation universality class. However, a different critical behavior is found at the point where the directed percolation phase boundaries meet. The values of the critical exponents calculated at the bicritical point are in good agreement with the exponents corresponding to the parity-conserving universality class. Since the adsorption-reaction processes does not lead to a non-trivial local parity-conserving dynamics, this result confirms that the twofold symmetry between absorbing states plays a relevant role in determining the universality class. The value of the exponent $\delta_2$, which characterizes the fluctuations of an interface at the bicritical point, supports the Bassler-Brown's conjecture which states that this is a new exponent in the parity-conserving universality class.

Critical phenomena of nonequilibrium dynamical systems with two absorbing states

We study nonequilibrium dynamical models with two absorbing states: interacting monomer-dimer models, probabilistic cellular automata models, nonequilibrium kinetic Ising models. These models exhibit a continuous phase transition from an active phase into an absorbing phase which belongs to the universality class of the models with the parity conservation. However, when we break the symmetry between the absorbing states by introducing a symmetry-breaking field, Monte Carlo simulations show that the system goes back to the conventional directed percolation universality class. In terms of domain wall language, the parity conservation is not affected by the presence of the symmetry-breaking field. So the symmetry between the absorbing states rather than the conservation laws plays an essential role in determining the universality class. We also perform Monte Carlo simulations for the various interface dynamics between different absorbing states, which yield new universal dynamic exponents. With the symmetry-breaking field, the interface moves, in average, with a constant velocity in the direction of the unpreferred absorbing state and the dynamic scaling exponents apparently assume trivial values. However, we find that the hyperscaling relation for the directed percolation universality class is restored if one focuses on the dynamics of the interface on the side of the preferred absorbing state only.

Phase Structure of Systems with Infinite Numbers of Absorbing States

Critical properties of systems exhibiting phase transitions into phases with infinite numbers of absorbing states are studied. We analyze a non-Markovian Langevin equation recently proposed to describe the critical behavior of such systems, and also introduce and study a non-Markovian discrete model, which is argued to present the same critical features. On the basis of mean-field analysis, Monte Carlo simulations, and theoretical arguments, we conclude that the phenomenology of the non-Markovian models closely parallels that of systems with many absorbing states in one and two dimensions. The “bulk” or “static” critical properties of these systems fall in the directed percolation (DP) universality class. By contrast, the critical properties associated with the spread of an initially localized seed exhibit a more complex behavior: Depending on parameter values they can, both in one and two dimensions, fall either in the dynamical percolation or DP universality class, or else exhibit apparently nonuniversal exponents. In contrast to previous results, however, the nonuniversal exponents in 2D are found to satisfy a scaling law which implies that a particular linear combination of them is universal and assumes DP values. These results demonstrate the efficacy of the non-Markovian approach for understanding systems with many absorbing states, which are difficult to analyze in their original microscopic formulation.

Damage spreading in the Ising model

We present two new results regarding damage spreading in ferromagnetic Ising models. First, we show that a damage spreading transition can occur in an Ising chain that evolves in contact with a thermal reservoir. Damage heals at low temperature and spreads for high T. The dynamic rules for the system's evolution for which such a transition is observed are as legitimate as the conventional rules (Glauber, Metropolis, heat bath). Our second result is that such transitions are not always in the directed percolation universality class.

Damage spreading in a two-dimensional Potts ferromagnet in an external field

We study, using the damage spreading method and within the heat bath dynamics, the dynamical behaviour of the square lattice three-state Potts ferromagnet subjected to an external field. In the zero field case, we verify that this very simple model presents, besides the transition consistent with the equilibrium one, a new dynamical chaotic phase with unusual features. Although the unexpected transition occurs, within the error bars, at the static Ising critical temperature, it is in the directed percolation universality class. We observe that the application of a uniform magnetic field does not destroy any of the two transitions, while the unexpected one is annihilated by a new kind of field which plays the role of a conjugate field to the Hamming distance.

Theory of Branching and Annihilating Random Walks.

A systematic theory for the diffusion--limited reaction processes $A + A \to 0$ and $A \to (m+1) A$ is developed. Fluctuations are taken into account via the field--theoretic dynamical renormalization group. For $m$ even the mean field rate equation, which predicts only an active phase, remains qualitatively correct near $d_c = 2$ dimensions; but below $d_c' \approx 4/3$ a nontrivial transition to an inactive phase governed by power law behavior appears. For $m$ odd there is a dynamic phase transition for any $d \leq 2$ which is described by the directed percolation universality class.

BRANCH ANNIHILATION RANDOM WALKS IN ENERGETIC DISORDERED STRUCTURE AND DIRECTED POLYMER

The problem of the branch annihilation random walks (BAW) in energetic disordered structure is introduced and studied on the checkerboard lattice of one spatial and one temporal dimensions. The energy disorder we considered is the one used in the problem of directed polymer (DPO) at zero temperature including the external field F. The evolution rule of BAW is assigned to relate to the energy disorder. It is found that depending on the value of F, a variety of phases appear, and the crossover from the DPO state to the directed percolation (DPE) state occurs. When F > 0, the problem is reduced to the DPO problem at zero temperature. However, for F < 0, there exist three states, the inactive state for F c < F < 0, the active state for − 1 < F < F c , and the non-hopping state for F < -1, where F c is the threshold value from the inactive state to the active state. The phase transition at F c turns out to be in the DPE universality class. The robustness of the DPE universality is discussed by the Harris criterion.

Phase diagram of branched polymer collapse.

The phase diagram of the collapse of a two-dimensional infinite branched polymer interacting with the solvent and with itself through contact interactions is studied from the $q\to 1$ limit of an extension of the $q-$ states Potts model. Exact solution on the Bethe lattice and Migdal-Kadanoff renormalization group calculations show that there is a line of $\theta$ transitions from the extended to a single compact phase. The $\theta$ line, governed by three different fixed points, consists of two lines of extended--compact transitions which are in different universality classes and meet in a multicritical point. On the other hand, directed branched polymers are shown to be completely determined by the strongly embedded case and there is a single $\theta$ transition which is in the directed percolation universality class.

Directed surfaces in disordered media.

The critical exponents for a class of one-dimensional models of interface depinning in disordered media can be calculated through a mapping onto directed percolation. In higher dimensions these models give rise to directed surfaces, which do not belong to the directed percolation universality class. We formulate a scaling theory of directed surfaces, and calculate critical exponents numerically, using a cellular automaton that locates the directed surfaces without making reference to the dynamics of the underlying interface growth models.

Critical behavior of an absorbing phase transition in an interacting monomer-dimer model

We study a monomer-dimer model with repulsive interactions between the same species in one dimension. With infinitely strong interactions the model exhibits a continuous transition from a reactive phase to an inactive phase with two equivalent absorbing states. Static and dynamic Monte Carlo simulations show that the critical behavior at the transition is different from the conventional directed percolation universality class but is consistent with that of the models with the mass conservation of modulo 2. The values of static and dynamic critical exponents are compared with those of other models. We also show that the directed percolation universality class is recovered when a symmetry-breaking field is introduced.

Dynamic scaling behavior of an interacting monomer-dimer model.

We study the dynamic scaling behavior of a monomer-dimer model with repulsive interactions between the same species in one dimension. With infinitely strong interactions the model exhibits a continuous transition from a reactive phase to an inactive phase with two equivalent absorbing states. This transition does not belong to the conventional directed percolation universality class. The values of dynamic scaling exponents are estimated by Monte Carlo simulations for two distinct initial configurations, one near an absorbing state and the other with an interface between two different absorbing states. We confirm that the critical behavior is consistent with that of the models with the mass conservation of modulo 2.

A parity conserving model with spontaneous annihilation

We propose a new model with the parity conservation of the total number of particles and show that this model belongs to the directed percolation universality class.