Electronic Journal of Probability

Convergence Results and Sharp Estimates for the Voter Model Interfaces

Samir Belhaouari, Thomas Mountford, Rongfeng Sun, and Glauco Valle

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82B24: Interface problems; diffusion-limited aggregation 82B41: Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41] 60F17: Functional limit theorems; invariance principles

voter model interface coalescing random walks Brownian web invariance principle

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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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