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arXiv:2512.02241 · Physical review. E

High-precision simulations of the parity-conserving directed-percolation universality class in 1 + 1 dimensions

Peter Grassberger

Published Jul 1, 2026• 0 citations
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Abstract

Next to the directed-percolation (DP) universality class, parity-conserving directed-percolation (pcDP; also called parity-conserving branching-annihilating random walks) is the second-most important model with an absorbing state transition. Its distinction from ordinary DP is that particle number is conserved modulo 2, which implies that there are two distinct sectors in systems with a finite initial number of particles: Realizations with even and odd particle numbers show different scaling behaviors, and systems in the odd sector cannot die. An intriguing feature of pcDP is that some of its critical exponents seem to be very simple rational numbers. The most prominent is the one describing the average number of particles (or active sites) in the even sector, which is asymptotically constant. In contrast, the dynamical critical exponent (which is the same in both sectors) seems not close to any simple rational. Finally, the order parameter exponent β (which is also the same in both sectors) is, according to the most precise previous simulations, rather close to 1, but incompatible with it. We present high statistics simulations which clarify this situation, and which indicate several other intriguing properties of pcDP clusters. In particular, we find that all exponents which were close to rationals are even closer, and β=1.000 with the error in the next digit.

DOI
10.1103/hw73-7vjg

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Next to the directed-percolation (DP) universality class, parity-conserving directed-percolation (pcDP; also called parity-conserving branching-annihilating random walks) is the second-most important model with an absorbing state transition. Its distinction from ordinary DP is that particle number is conserved modulo 2, which implies that there are two distinct sectors in systems with a finite initial number of particles: Realizations with even and odd particle numbers show different scaling behaviors, and systems in the odd sector cannot die. An intriguing feature of pcDP is that some of its critical exponents seem to be very simple rational numbers. The most prominent is the one describing the average number of particles (or active sites) in the even sector, which is asymptotically constant. In contrast, the dynamical critical exponent (which is the same in both sectors) seems not close to any simple rational. Finally, the order parameter exponent β (which is also the same in both sectors) is, according to the most precise previous simulations, rather close to 1, but incompatible with it. We present high statistics simulations which clarify this situation, and which indicate several other intriguing properties of pcDP clusters. In particular, we find that all exponents which were close to rationals are even closer, and β=1.000 with the error in the next digit.
Absorbing phase transitions in restricted exclusion processes are characterized by simple integer exponents. We show that this critical behaviour flows to the directed percolation (DP) universality class when particle conservation is broken suitably. The same transition, when studied using the average density as the controlling parameter, yields critical exponents quite different from DP; we argue that these exponents are actually related to DP by a scaling factor 1/ β DP . These conclusions also apply to conserved lattice gas in one dimension.
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 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.
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.
In this dissertation we address the problem of second order phase transitions with absorbing states. We analyze the critical behavior of a one dimensional model simulated on one dimensional chains. The sites on the linear chain can be multiple occupied by particles that annhilates each other with a finite probability. It also presents parity-conserving branching with an even number of offsprings. The number of particles are constant modulo 2. We determine the threshold of the phase transition between the statistically stationary active state and the absorbing state. From steady-state simulations and a finite-size scaling analysis, we calculate the order-parameter, order-parameter fluctuations, and special correlation length critical exponents. We also followed the short-time critical relaxation to determine some dynamical critical exponents. The hyperscaling relation is checked for both sets of stationary and dynamical critical exponents which we found to be consistent with the BARW-PC universality class.
The pair contact process with diffusion (PCPD) with modulo 2 conservation (PCPD2) [ 2A-->4A , 2A-->0 ] is studied in one dimension, focused on the crossover to other well established universality classes: the directed Ising (DI) and the directed percolation (DP). First, we show that the PCPD2 shares the critical behaviors with the PCPD, both with and without directional bias. Second, the crossover from the PCPD2 to the DI is studied by including a parity-conserving single-particle process (A-->3A) . We find the crossover exponent 1/varphi_{1}=0.57(3) , which is argued to be identical to that of the PCPD-to-DP crossover by adding A-->2A . This suggests that the PCPD universality class has a well-defined fixed point distinct from the DP. Third, we study the crossover from a hybrid-type reaction-diffusion process belonging to the DP [ 3A-->5A , 2A-->0 ] to the DI by adding A-->3A . We find 1/varphi_{2}=0.73(4) for the DP-to-DI crossover. The inequality of varphi_{1} and varphi_{2} further supports the non-DP nature of the PCPD scaling. Finally, we introduce a symmetry-breaking field in the dual spin language to study the crossover from the PCPD2 to the DP. We find 1/varphi_{3}=1.23(10) , which is associated with a new independent route from the PCPD to the DP.