## Electronic Journal of Probability

### Convergence Results and Sharp Estimates for the Voter Model Interfaces

#### Abstract

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.

#### Article information

Source
Electron. J. Probab., Volume 11 (2006), paper no. 30, 768-801.

Dates
Accepted: 29 August 2006
First available in Project Euclid: 31 May 2016

https://projecteuclid.org/euclid.ejp/1464730565

Digital Object Identifier
doi:10.1214/EJP.v11-349

Mathematical Reviews number (MathSciNet)
MR2242663

Zentralblatt MATH identifier
1113.60092

Rights

#### Citation

Belhaouari, Samir; Mountford, Thomas; Sun, Rongfeng; Valle, Glauco. Convergence Results and Sharp Estimates for the Voter Model Interfaces. Electron. J. Probab. 11 (2006), paper no. 30, 768--801. doi:10.1214/EJP.v11-349. https://projecteuclid.org/euclid.ejp/1464730565

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