Passage-time moments and hybrid zones for the exclusion-voter model

Iain M. MacPhee, Mikhail V. Menshikov, Stanislav Volkov, and Andrew R. Wade

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We study the non-equilibrium dynamics of a one-dimensional interacting particle system that is a mixture of the voter model and the exclusion process. With the process started from a finite perturbation of the ground state Heaviside configuration consisting of 1’s to the left of the origin and 0’s elsewhere, we study the relaxation time τ, that is, the first hitting time of the ground state configuration (up to translation). We give conditions for τ to be finite and for certain moments of τ to be finite or infinite, and prove a result that approaches a conjecture of Belitsky et al. (Bernoulli 7 (2001) 119–144). Ours are the first non-existence-of-moments results for τ for the mixture model. Moreover, we give almost sure asymptotics for the evolution of the size of the hybrid (disordered) region. Most of our results pertain to the discrete-time setting, but several transfer to continuous-time. As well as the mixture process, some of our results also cover pure exclusion. We state several significant open problems.

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Bernoulli Volume 16, Number 4 (2010), 1312-1342.

First available in Project Euclid: 18 November 2010

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almost-sure bounds exclusion process hybrid zone Lyapunov functions passage-time moments voter model


MacPhee, Iain M.; Menshikov, Mikhail V.; Volkov, Stanislav; Wade, Andrew R. Passage-time moments and hybrid zones for the exclusion-voter model. Bernoulli 16 (2010), no. 4, 1312--1342. doi:10.3150/09-BEJ243.

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  • [1] Andjel, E., Ferrari, P.A. and Siqueira, A. (2004). Law of large numbers for the simple exclusion process. Stochastic Process. Appl. 113 217–233.
  • [2] Aspandiiarov, S. and Iasnogorodski, R. (1997). Tails of passage-time for non-negative stochastic processes and an application to stochastic processes with boundary reflection in a wedge. Stochastic Process. Appl. 66 115–145.
  • [3] Aspandiiarov, S. and Iasnogorodski, R. (1999). General criteria of integrability of functions of passage-times for nonnegative stochastic processes and their applications. Theory Probab. Appl. 43 343–369.
  • [4] Aspandiiarov, S., Iasnogorodski, R. and Menshikov, M. (1996). Passage-time moments for nonnegative stochastic processes and an application to reflected random walks in a quadrant. Ann. Probab. 24 932–960.
  • [5] Belhaouari, S., Mountford, T., Sun, R. and Valle, G. (2006). Convergence results and sharp estimates for the voter model interfaces. Electron. J. Probab. 11 768–801.
  • [6] Belhaouari, S., Mountford, T. and Valle, G. (2007). Tightness for the interfaces of one-dimensional voter models. Proc. London Math. Soc. (3) 94 421–442.
  • [7] Belitsky, V., Ferrari, P.A., Menshikov, M.V. and Popov, S.Yu. (2001). A mixture of the exclusion process and the voter model. Bernoulli 7 119–144.
  • [8] Clifford, P. and Sudbury, A. (1973). A model for spatial conflict. Biometrika 60 581–588.
  • [9] Cox, J.T. and Durrett, R. (1995). Hybrid zones and voter model interfaces. Bernoulli 1 343–370.
  • [10] Durrett, R. and Zähle, I. (2007). On the width of hybrid zones. Stochastic Process. Appl. 117 1751–1763.
  • [11] Fayolle, G., Malyshev, V.A. and Menshikov, M.V. (1995). Topics in the Constructive Theory of Countable Markov Chains. Cambridge: Cambridge Univ. Press.
  • [12] Foster, F.G. (1953). On the stochastic matrices associated with certain queuing processes. Ann. Math. Statist. 24 355–360.
  • [13] Holley, R.A. and Liggett, T.M. (1975). Ergodic theorems for weakly interacting infinite systems and the voter model. Ann. Probab. 3 643–663.
  • [14] Jung, P. (2005). The noisy voter–exclusion process. Stochastic Process. Appl. 115 1979–2005.
  • [15] Lamperti, J. (1963). Criteria for stochastic processes II: Passage-time moments. J. Math. Anal. Appl. 7 127–145.
  • [16] Liggett, T.M. (1985). Interacting Particle Systems. New York: Springer.
  • [17] Liggett, T.M. (1999). Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Berlin: Springer.
  • [18] Menshikov, M.V., Vachkovskaia, M. and Wade, A.R. (2008). Asymptotic behaviour of randomly reflecting billiards in unbounded tubular domains. J. Stat. Phys. 132 1097–1133.
  • [19] Rost, H. (1981). Non-equilibrium behaviour of a many particle process: Density profile and local equilibria. Z. Wahrsch. Verw. Gebiete 58 41–53.
  • [20] Sturm, A. and Swart, J.M. (2008). Tightness of voter model interfaces. Electron. Comm. Probab. 13 165–174.