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