Convergence Results and Sharp Estimates for the Voter Model Interfaces

Samir Belhaouari (EPFL)
Thomas Mountford (EPFL)
Rongfeng Sun (EURANDOM)
Glauco Valle (EPFL / DME-IM-UFRJ)


We study the evolution of the interface for the one-dimensional voter model. We show that if the random walk kernel associated with the voter model has finite $\gamma$-th moment for some $\gamma > 3$, then the evolution of the interface boundaries converge weakly to a Brownian motion under diffusive scaling. This extends recent work of Newman, Ravishankar and Sun. Our result is optimal in the sense that finite $\gamma$-th moment is necessary for this convergence for all $\gamma \in (0,3)$. We also obtain relatively sharp estimates for the tail distribution of the size of the equilibrium interface, extending earlier results of Cox and Durrett, and Belhaouari, Mountford and Valle.

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Pages: 768-801

Publication Date: August 29, 2006

DOI: 10.1214/EJP.v11-349


  1. S. Belhaouari, T. Mountford, G. Valle, Tightness of the interface for one-dimensional voter models, Preprint . Math. Review number not available.
  2. J. van den Berg, H. Kesten, Randomly coalescing random walk in dimension $\geq3$, In and out of equilibrium (Mambucaba, 2000), 1--45, Progr. Probab., 51, Birkh\"auser Boston, 2002. MR1901947.
  3. P. Billingsley, Convergence of Probability Measures, 2nd edition. John Wiley & Sons, 1999. MR1700749.
  4. J. T. Cox, R. Durrett, Hybrid zones and voter model interfaces, Bernoulli, 1 (1995), 343-370. MR1369166.
  5. L. R. G. Fontes, M. Isopi, C. M. Newman, K. Ravishankar, The Brownian web, Proc. Nat. Acad. Sciences 99 (2002), 15888-15893. MR1944976.
  6. L. R. G. Fontes, M. Isopi, C. M. Newman, K. Ravishankar, The Brownian web: characterization and convergence, Annals of Probability 32 (2004), 2857-2883. MR2094432.
  7. T. M. Liggett, Interacting Particle Systems, Springer-Verlag, 1985. MR0776231.
  8. S. V. Nagaev, Large deviations of sums of independent random variables, Ann. Prob. 7, 745-789, 1979. MR0542129.
  9. C. M. Newman, K. Ravishankar, R. Sun, Convergence of coalescing nonsimple random walks to the Brownian web, Electronic Journal of Probability 10, Paper 2, 2005. MR2120239.
  10. F. Spitzer, Principles of Random Walk, 2nd edition, Springer-Verlag, 1976. MR0388547.
  11. R. Sun, Ph.D. Thesis, Courant Institute of Mathematical Sciences, New York University, 2005, arxiv:math.PR/0501141. Math. Review number not available.

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