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Convergence results and sharp estimates for the voter model interfaces. (English) Zbl 1113.60092
Summary: 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 $\lambda$th moment for some $\lambda>3$, then the evolution of the interface boundaries converge weakly to a Brownian motion under diffusive scaling. This extends recent work of {\it C. M. Newman, K. Ravishankar} and {\it R. Sun} [Electron. J. Probab. 10, Paper No. 2, 21--60 (2005; Zbl 1067.60099)]. Our result is optimal in the sense that finite $\lambda$th moment is necessary for this convergence for all $\lambda\in (0,3)$. We also obtain relatively sharp estimates for the tail distribution of the size of the equilibrium interface, extending earlier results of {\it J. T. Cox} and {\it R. Durrett} [Bernoulli 1, No. 4, 343--370 (1995; Zbl 0849.60088)] and {\it S. Belhaouari, T. Mountford} and {\it G. Valle} [Proc. Lond. Math. Soc. (3) 94, No. 2, 421--442 (2007; Zbl 1112.60074)].

60K35Interacting random processes; statistical mechanics type models; percolation theory
60F17Functional limit theorems; invariance principles
82B24Interface problems; diffusion-limited aggregation (equilibrium statistical mechanics)
82B41Random walks, random surfaces, lattice animals, etc. (statistical mechanics)
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