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

Peter Grassberger

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Phys. Rev. E 114, 014150 – Published 29 July, 2026

DOI: https://doi.org/10.1103/hw73-7vjg

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.

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References (30)

  1. P. Grassberger, F. Krause, and T. von der Twer, A new type of kinetic critical phenomenon, J. Phys. A: Math. Gen. 17, L105 (1984).
  2. P. Grassberger, Some further results on a kinetic critical phenomenon, J. Phys. A: Math. Gen. 22, L1103 (1989).
  3. H. Takayasu and A. Y. Tretyakov, Extinction, survival, and dynamical phase transition of branching annihilating random walk, Phys. Rev. Lett. 68, 3060 (1992).
  4. I. Jensen, Critical exponents for branching annihilating random walks with an even number of offspring, Phys. Rev. E 50, 3623 (1994).
  5. D. Zhong and D. ben-Avraham, Universality class of two-offspring branching annihilating random walks, Phys. Lett. A 209, 333 (1995).
  6. J. L. Cardy and U. C. Täuber, Field theory of branching and annihilating random walks, J. Stat. Phys. 90, 1 (1998).
  7. N. Menyhárd, One-dimensional non-equilibrium kinetic Ising models with branching annihilating random walk, J. Phys. A: Math. & General 27, 6139 (1994).
  8. N. Menyhárd and G. Ódor, Nonequilibrium kinetic Ising models: Phase transitions and universality classes in one dimension, Braz. J. Phys. 30, 113 (2000).
  9. H. Hinrichsen, Stochastic lattice models with several absorbing states, Phys. Rev. E 55, 219 (1997).
  10. H. Ha, D. A. Huse, and R. Samajdar, Absorbing state transitions with discrete symmetries, arXiv:2502.08702.
  11. H.-K. Janssen, On the nonequilibrium phase transition in reaction-diffusion systems with an absorbing stationary state, Z. Phys. B 42, 151 (1981).
  12. P. Grassberger, On phase transitions in Schlögl's 2nd model, Z. Phys. B 47, 365 (1982).
  13. S.-C. Park, High-precision estimate of the critical exponents for the directed Ising universality class, J. Korean Phys. Soc. 62, 469 (2013).
  14. P. Grassberger, On the continuum time limit of reaction-diffusion systems, Europhys. Lett. 103, 50009 (2013).
  15. P. Nolin, W. Qian, X. Sun, and Z. Zhuang, Backbone exponent for two-dimensional percolation, Phys. Rev. Lett. 134, 117101 (2025).
  16. K. Mussawisade, J. Santos, and G. Schütz, Branching-annihilating random walks in one dimension: Some exact results, J. Phys. A: Math. Gen. 31, 4381 (1998).
  17. S.-C. Park, Order-parameter critical exponent of absorbing phase transitions in one-dimensional systems with two symmetric absorbing states, Phys. Rev. E 101, 052114 (2020).
  18. A. Sudbury, The branching annihilating process: An interacting particle system, Ann. Probab. 18, 581 (1990).
  19. I. Jensen, Series expansions for the percolation probability of a generalized Domany-Kinzel cellular automaton, J. Phys. A: Math. Gen. 30, 8471 (1997).
  20. D. R. Cox, Renewal Theory (Chapman and Hall, London, 1962).
  21. C. Godrèche and J. Luck, Statistics of the occupation time of renewal processes, J. Stat. Phys. 104, 489 (2001).
  22. C. Godrèche, Comment on ‘fluctuation-dominated phase ordering at a mixed order transition’, J. Phys. A: Math. Theor. 54, 038001 (2021).
  23. P. Grassberger, D. Dhar, and P. K. Mohanty, Oslo model, hyperuniformity, and the quenched Edwards-Wilkinson model, Phys. Rev. E 94, 042314 (2016).
  24. S. Caracciolo, A. Guttmann, I. Jensen, A. Pelissetto, A. Rogers, and A. Sokal, Correction-to-scaling exponents for two-dimensional self-avoiding walks, J. Stat. Phys. 120, 1037 (2005).
  25. P. Grassberger, Conductivity exponent and backbone dimension in 2-D percolation, Physica A 262, 251 (1999).
  26. P. Grassberger and Y.-C. Zhang, “Self-organized” formulation of standard percolation phenomena, Physica A 224, 169 (1996).
  27. H. Ballesteros, L. Fernández, V. Martín-Mayor, A. Muñoz, G. Parisi, J. Ruiz-Lorenzo, et al., Measures of critical exponents in the four-dimensional site percolation, Phys. Lett. B 400, 346 (1997).
  28. R. Dickman, Reweighting in nonequilibrium simulations, Phys. Rev. E 60, R2441 (1999).
  29. M. E. J. Newman and R. M. Ziff, Fast Monte Carlo algorithm for site or bond percolation, Phys. Rev. E 64, 016706 (2001).
  30. P. Grassberger, Critical percolation in high dimensions, Phys. Rev. E 67, 036101 (2003).

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Phys. Rev. E 114, 014150– Published 29 July, 2026
  • Received 16 December 2025
  • Accepted 1 July 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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