Abstract
The rebellious voter model, introduced by Sturm and Swart (2008), is a variation of the standard, one-dimensional voter model, in which types that are locally in the minority have an advantage. It is related, both through duality and through the evolution of its interfaces, to a system of branching annihilating random walks that is believed to belong to the ‘parity-conservation’ universality class. This paper presents numerical data for the rebellious voter model and for a closely related one-sided version of the model. Both models appear to exhibit a phase transition between noncoexistence and coexistence as the advantage for minority types is increased. For the one-sided model (but not for the original, two-sided rebellious voter model), it appears that the critical point is exactly a half and two important functions of the process are given by simple, explicit formulas, a fact for which we have no explanation.
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Blath, J., Etheridge, A., Meredith, M.: Coexistence in locally regulated competing populations and survival of branching annihilating random walk. Ann. Appl. Probab. 17(5–6), 1474–1507 (2007)
Canet, L., Chaté, H., Delmotte, B., Dornic, I., Muñoz, M.A.: Nonperturbative fixed point in a nonequilibrium phase transition. Phys. Rev. Lett. 95, 100601 (2005)
Cox, J.T., Durrett, R.: Hybrid zones and voter model interfaces. Bernoulli 1(4), 343–370 (1995)
Cox, J.T., Merle, M., Perkins, E.A.: Co-existence in a two-dimensional Lotka-Volterra model. Preprint (2009)
Cox, J.T., Perkins, E.A.: Survival and coexistence in stochastic spatial Lotka-Volterra models. Probab. Theory Relat. Fields 139(1–2), 89–142 (2007)
Clifford, P., Sudbury, A.: A model for spatial conflict. Biometrika 60, 581–588 (1973)
Cardy, J.L., Täuber, U.C.: Field theory of branching and annihilating random walks. J. Stat. Phys. 90, 1–56 (1998)
Grassberger, P., Krause, F., Von der Twer, T.: A new type of kinetic critical phenomenon. J. Phys. A, Math. Gen. 17, 105–109 (1984)
Griffeath, D.: Additive and Cancellative Interacting Particle Systems. Lecture Notes in Math., vol. 724. Springer, Berlin (1979)
Hinrichsen, H.: Nonequilibrium critical phenomena and phase transitions into absorbing states. Adv. Phys. 49(7), 815–958 (2000)
Holley, R., Liggett, T.M.: Ergodic theorems for weakly interacting systems and the voter model. Ann. Probab. 3, 643–663 (1975)
Inui, N., Tretyakov, A.Yu.: Critical behavior of the contact process with parity conservation. Phys. Rev. Lett. 80(23), 5148–5151 (1998)
Jensen, I.: Critical exponents for branching annihilating random walks with an even number of offspring. Phys. Rev. E 50(5), 3623–3633 (1994)
Neuhauser, C., Pacala, S.W.: An explicitly spatial version of the Lotka-Volterra model with interspecific competition. Ann. Appl. Probab. 9(4), 1226–1259 (1999)
Ódor, G., Szolnoki, A.: Cluster mean field study of the parity-conserving phase transition. Phys. Rev. E 71(6), 066128 (2005)
Ódor, G., Menyhárd, N.: Critical behavior of an even-offspringed branching and annihilating random-walk cellular automaton with spatial disorder. Phys. Rev. E 73, 036130 (2006)
Sturm, A., Swart, J.M.: Voter models with heterozygosity selection. Ann. Appl. Probab. 18(1), 59–99 (2008)
Sturm, A., Swart, J.M.: Tightness of voter model interfaces. Electron. Commun. Probab. 13(16), 165–174 (2008)
Sudbury, A.: The branching annihilating process: an interacting particle system. Ann. Probab. 18, 581–601 (1990)
Takayasu, H., Tretyakov, A.Yu.: Extinction, survival, and dynamical phase transition of branching annihilating random walk. Phys. Rev. Lett. 68(20), 3060–3063 (1992)
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Swart, J.M., Vrbenský, K. Numerical Analysis of the Rebellious Voter Model. J Stat Phys 140, 873–899 (2010). https://doi.org/10.1007/s10955-010-0021-x
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DOI: https://doi.org/10.1007/s10955-010-0021-x