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Universal and nonuniversal features of the generalized voter class for ordering dynamics in two dimensions

Claudio Castellano1,2 and Romualdo Pastor-Satorras3

  • 1Istituto dei Sistemi Complessi (ISC-CNR), via dei Taurini 19, I-00185 Roma, Italy
  • 2Dipartimento di Fisica, “Sapienza” Università di Roma, Piazzale A. Moro 2, I-00185 Roma, Italy
  • 3Departament de Física i Enginyeria Nuclear, Universitat Politècnica de Catalunya, Campus Nord B4, E-08034 Barcelona, Spain

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Phys. Rev. E 86, 051123 – Published 21 November, 2012

DOI: https://doi.org/10.1103/PhysRevE.86.051123

Abstract

By considering three different spin models belonging to the generalized voter class for ordering dynamics in two dimensions [Dornic et al., Phys. Rev. Lett. 87, 045701 (2001)], we show that they behave differently from the linear voter model when the initial configuration is an unbalanced mixture of up and down spins. In particular, we show that for nonlinear voter models the exit probability (probability to end with all spins up when starting with an initial fraction x of them) assumes a nontrivial shape. This is the first time a nontrivial exit probability is observed in two-dimensional systems. The change is traced back to the strong nonconservation of the average magnetization during the early stages of dynamics. Also the time needed to reach the final consensus state TN(x) has an anomalous nonuniversal dependence on x.

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References (29)

  1. P. Clifford and A. Sudbury, Biometrika 60, 581 (1973).
  2. R. A. Holley and T. M. Liggett, Ann. Probab. 3, 643 (1975).
  3. A. J. Bray, Adv. Phys. 43, 357 (1994).
  4. P. Krapivsky, S. Redner, and E. Ben-Naim, A Kinetic View of Statistical Physics (Cambridge University Press, Cambridge England, 2010).
  5. T. M. Liggett, Stochastic Interacting Particle Systems: Contact, Voter, and Exclusion Processes (Springer-Verlag, New York, 1999).
  6. L. Frachebourg and P. L. Krapivsky, Phys. Rev. E 53, R3009 (1996).
  7. C. Castellano, S. Fortunato, and V. Loreto, Rev. Mod. Phys. 81, 591 (2009).
  8. J. F. Crow and M. Kimura, An Introduction to Population Genetics Theory (Harper & Row, New York, 1970).
  9. S. Hubbell, The Unified Neutral Theory of Biodiversity and Biogeography, Monographs in Population Biology (Princeton University Press, Princeton, NJ, 2001).
  10. R. A. Blythe, J. Stat. Mech.: Theory Exp. (2009) P02059.
  11. I. Dornic, H. Chaté, J. Chave, and H. Hinrichsen, Phys. Rev. Lett. 87, 045701 (2001).
  12. R. A. Blythe, J. Phys. A 43, 385003 (2010).
  13. M. de Oliveira, J. Mendes, and M. Santos, J. Phys. A 26, 2317 (1993).
  14. J.-M. Drouffe and C. Godrèche, J. Phys. A 32, 249 (1999).
  15. J. Molofsky, R. Durrett, J. Dushoff, D. Griffeath, and S. Levin, Theoretical Population Biology 55, 270 (1999).
  16. O. Al Hammal, H. Chaté, I. Dornic, and M. A. Muñoz, Phys. Rev. Lett. 94, 230601 (2005).
  17. F. Vázquez and C. López, Phys. Rev. E 78, 061127 (2008).
  18. L. Canet, H. Chaté, B. Delamotte, I. Dornic, and M. A. Muñoz, Phys. Rev. Lett. 95, 100601 (2005).
  19. M. Droz, A. L. Ferreira, and A. Lipowski, Phys. Rev. E 67, 056108 (2003).
  20. J. Marro and R. Dickman, Nonequilibrium Phase Transitions in Lattice Models (Cambridge University Press, Cambridge, England, 1999).
  21. C. Castellano, M. A. Muñoz, and R. Pastor-Satorras, Phys. Rev. E 80, 041129 (2009).
  22. H. Kaya, A. Kabakçioǧlu, and A. Erzan, Phys. Rev. E 61, 1102 (2000).
  23. F. Corberi, E. Lippiello, and M. Zannetti, Phys. Rev. E 78, 011109 (2008).
  24. C. Gardiner, Stochastic Methods: A Handbook for the Natural and Social Sciences, 4th ed. (Springer-Verlag, Berlin, 2010).
  25. C. Castellano and R. Pastor-Satorras, Phys. Rev. E 83, 016113 (2011).
  26. V. Sood and S. Redner, Phys. Rev. Lett. 94, 178701 (2005).
  27. N. Masuda, N. Gibert, and S. Redner, Phys. Rev. E 82, 010103 (2010).
  28. R. Lambiotte and S. Redner, Europhys. Lett. 82, 18007 (2008).
  29. F. Slanina, K. Sznajd-Weron, and P. Przybyła, Europhys. Lett. 82, 18006 (2008).

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Phys. Rev. E 86, 051123– Published 21 November, 2012
  • Received 18 July 2012

©2012 American Physical Society

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