Electronic Communications in Probability

One-dimensional Voter Model Interface Revisited

Siva Athreya and Rongfeng Sun

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We consider the voter model on $\mathbb{Z}$, starting with all 1's to the left of the origin and all $0$'s to the right of the origin. It is known that if the associated random walk kernel $p$ has zero mean and a finite r-th moment for any $r>3$, then the evolution of the boundaries of the interface region between 1's and 0's converge in distribution to a standard Brownian motion $(B_t)_{t>0}$ under diffusive scaling of space and time. This convergence fails when $p$ has an infinite $r$-th moment for any $r<3$, due to the loss of tightness caused by a few isolated $1$'s appearing deep within the regions of all $0$'s (and vice versa) at exceptional times. In this note, we show that as long as $p$ has a finite second moment, the measure-valued process induced by the rescaled voter model configuration is tight, and converges weakly to the measure-valued process $1_{x< B_t} dx$, $t>0$.

Article information

Electron. Commun. Probab., Volume 16 (2011), paper no. 70, 792-800.

Accepted: 7 December 2011
First available in Project Euclid: 7 June 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: 82C22: Interacting particle systems [See also 60K35] 82C24: Interface problems; diffusion-limited aggregation 60F17: Functional limit theorems; invariance principles

voter model interface measure-valued process tightness

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Athreya, Siva; Sun, Rongfeng. One-dimensional Voter Model Interface Revisited. Electron. Commun. Probab. 16 (2011), paper no. 70, 792--800. doi:10.1214/ECP.v16-1688. https://projecteuclid.org/euclid.ecp/1465262026

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