## 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

### Citations:

Zbl 1067.60099; Zbl 0849.60088; Zbl 1112.60074
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