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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 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 \(\lambda\)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 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:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60F17 Functional limit theorems; invariance principles
82B24 Interface problems; diffusion-limited aggregation arising in equilibrium statistical mechanics
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
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