At nonequilibrium phase transitions into absorbing (trapped) states, it is well known that the directed percolation (DP) critical scaling is shared by two classes of models with a single (S) absorbing state and with infinitely many (IM) absorbing states. We study the crossover behavior in one dimension, arising from a considerable reduction of the number of absorbing states (typically from the IM-type to the S -type DP models) by following two different (excitatory or inhibitory) routes which make the auxiliary field density abruptly jump at the crossover. Along the excitatory route, the system becomes overly activated even for an infinitesimal perturbation and its crossover becomes discontinuous. Along the inhibitory route, we find a continuous crossover with universal crossover exponent phi approximately=1.78(6), which is argued to be equal to nu||, the relaxation time exponent of the DP universality class on a general footing. This conjecture is also confirmed in the case of the directed Ising (parity-conserving) class. Finally, we discuss the effect of diffusion on the IM-type models and suggest an argument why diffusive models with some hybrid-type reactions should belong to the DP class.
At nonequilibrium phase transitions into absorbing (trapped) states, it is well known that the directed percolation (DP) critical scaling is shared by two classes of models with a single (S) absorbing state and with infinitely many (IM) absorbing states. We study the crossover behavior in one dimension, arising from a considerable reduction of the number of absorbing states (typically from the IM-type to the S -type DP models) by following two different (excitatory or inhibitory) routes which make the auxiliary field density abruptly jump at the crossover. Along the excitatory route, the system becomes overly activated even for an infinitesimal perturbation and its crossover becomes discontinuous. Along the inhibitory route, we find a continuous crossover with universal crossover exponent phi approximately=1.78(6), which is argued to be equal to nu||, the relaxation time exponent of the DP universality class on a general footing. This conjecture is also confirmed in the case of the directed Ising (parity-conserving) class. Finally, we discuss the effect of diffusion on the IM-type models and suggest an argument why diffusive models with some hybrid-type reactions should belong to the DP class.
The scaling nature of absorbing critical phenomena is well understood for the directed percolation (DP) and the directed Ising (DI) systems. However, a full analysis of the crossover behavior is still lacking, which is of our interest in this study. In one dimension, we find three different routes from the DI to the DP classes by introducing a symmetry-breaking field (SB), breaking a modulo 2 conservation (CB), or making channels connecting two equivalent absorbing states (CC). Each route can be characterized by a crossover exponent, which is found numerically as phi=2.1+/-0.1 (SB), 4.6+/-0.2 (CB), and 2.9+/-0.1 (CC), respectively. The difference between the SB and CB crossover can be understood easily in the domain wall language, while the CC crossover involves an additional critical singularity in the auxiliary field density with the memory effect to identify itself independent.
The one-dimensional pair contact process with diffusion (PCPD), an interacting particle system with diffusion, pair annihilation, and creation by pairs, has defied consensus about the universality class to which it belongs. An argument by Hinrichsen [Physica A 361, 457 (2006)PHYADX0378-437110.1016/j.physa.2005.06.101] claims that freely diffusing particles in the PCPD should play the same role as frozen particles when it comes to the critical behavior. Therefore, the PCPD is claimed to have the same critical phenomena as a model with infinitely many absorbing states that belongs to the directed percolation (DP) universality class. To investigate if diffusing particles are really indistinguishable from frozen particles in the sense of the renormalization group, we study numerically a variation of the PCPD by introducing a nonorder field associated with infinitely many absorbing states. We find that a crossover from the PCPD to DP occurs due to the nonorder field. By studying a similar model, we exclude the possibility that the mere introduction of a nonorder field to one model can entail a nontrivial crossover to another model in the same universality class, thus we attribute the observed crossover to the difference of the universality class of the PCPD from the DP class.
Phase transitions from an active into an absorbing, inactive state are generically described by the critical exponents of directed percolation (DP), with upper critical dimension dc=4. In the framework of single-species reaction-diffusion systems, this universality class is realized by the combined processes $\stackrel{\ensuremath{\rightarrow}}{A}A+A,$ $A+\stackrel{\ensuremath{\rightarrow}}{A}A$, and $\stackrel{\ensuremath{\rightarrow}}{A}0$. We study a hierarchy of such DP processes for particle species A,B,…, unidirectionally coupled via the reactions $\stackrel{\ensuremath{\rightarrow}}{A}B,\dots{}$ (with rates ${\ensuremath{\mu}}_{\mathrm{AB}},\dots{})$. When the DP critical points at all levels coincide, multicritical behavior emerges, with density exponents ${\ensuremath{\beta}}_{i}$ which are markedly reduced at each hierarchy level i>2. This scenario can be understood on the basis of the mean-field rate equations, which yield ${\ensuremath{\beta}}_{i}{=1/2}^{i\ensuremath{-}1}$ at the multicritical point. Using field-theoretic renormalization-group techniques in $d=4\ensuremath{-}\ensuremath{\epsilon}$ dimensions, we identify a new crossover exponent $\ensuremath{\varphi}$, and compute $\ensuremath{\varphi}=1+O({\ensuremath{\epsilon}}^{2})$ in the multicritical regime (for small ${\ensuremath{\mu}}_{\mathrm{AB}})$ of the second hierarchy level. In the active phase, we calculate the fluctuation correction to the density exponent on the second hierarchy level, ${\ensuremath{\beta}}_{2}=1/2\ensuremath{-}\ensuremath{\epsilon}/8+O({\ensuremath{\epsilon}}^{2})$. Outside the multicritial region, we discuss the crossover to ordinary DP behavior, with the density exponent ${\ensuremath{\beta}}_{1}=1\ensuremath{-}\ensuremath{\epsilon}/6+O({\ensuremath{\epsilon}}^{2})$. Monte Carlo simulations are then employed to confirm the crossover scenario, and to determine the values for the new scaling exponents in dimensions d<3, including the critical initial slip exponent. Our theory is connected to specific classes of growth processes and to certain cellular automata, and the above ideas are also applied to unidirectionally coupled pair annihilation processes. We also discuss some technical as well as conceptual problems of the loop expansion, and suggest some possible interpretations of these difficulties.
We review the critical behavior of nonequilibrium systems, such as directed percolation (DP) and branching-annihilating random walks (BARW), which possess phase transitions into absorbing states. After reviewing the bulk scaling behavior of these models, we devote the main part of this review to analyzing the impact of walls on their critical behavior. We discuss the possible boundary universality classes for the DP and BARW models, which can be described by a general scaling theory which allows for two independent surface exponents in addition to the bulk critical exponents. Above the upper critical dimension d c , we review the use of mean field theories, whereas in the regime d<d c , where fluctuations are important, we examine the application of field theoretic methods. Of particular interest is the situation in d=1, which has been extensively investigated using numerical simulations and series expansions. Although DP and BARW fit into the same scaling theory, they can still show very different surface behavior: for DP some exponents are degenerate, a property not shared with the BARW model. Moreover, a "hidden" duality symmetry of BARW in d=1 is broken by the boundary and this relates exponents and boundary conditions in an intricate way.
We review the field theory approach to percolation processes. Specifically, we focus on the so-called simple and general epidemic processes that display continuous nonequilibrium active to absorbing state phase transitions whose asymptotic features are governed respectively by the directed (DP) and dynamic isotropic percolation (dIP) universality classes. We discuss the construction of a field theory representation for these Markovian stochastic processes based on fundamental phenomenological considerations, as well as from a specific microscopic reaction-diffusion model realization. Subsequently we explain how dynamic renormalization group (RG) methods can be applied to obtain the universal properties near the critical point in an expansion about the upper critical dimensions dc = 4 (DP) and 6 (dIP). We provide a detailed overview of results for critical exponents, scaling functions, crossover phenomena, finite-size scaling, and also briefly comment on the influence of long-range spreading, the presence of a boundary, multispecies generalizations, coupling of the order parameter to other conserved modes, and quenched disorder. Key words: Percolation, epidemic processes, directed percolation, dynamic isotropic percolation, active to absorbing phase transitions, renormalization group theory, dynamic critical phenomena, crossover
We review the critical behavior of nonequilibrium systems, such as directed percolation (DP) and branching-annihilating random walks (BARW), which possess phase transitions into absorbing states. After reviewing the bulk scaling behavior of these models, we devote the main part of this review to analyzing the impact of walls on their critical behavior. We discuss the possible boundary universality classes for the DP and BARW models, which can be described by a general scaling theory which allows for two independent surface exponents in addition to the bulk critical exponents. Above the upper critical dimension dc, we review the use of mean field theories, whereas in the regime d < dc, where fluctuations are important, we examine the application of field theoretic methods. Of particular interest is the situation in d = 1, which has been extensively investigated using numerical simulations and series expansions. Although DP and BARW fit into the same scaling theory, they can still show very different surface behavior: for DP some exponents are degenerate, a property not shared with the BARW model. Moreover, a “hidden ” duality symmetry of BARW in d = 1 is broken by the boundary and this relates exponents and boundary conditions in an intricate way.