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2011 One-dimensional Voter Model Interface Revisited
Siva Athreya, Rongfeng Sun
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Electron. Commun. Probab. 16: 792-800 (2011). DOI: 10.1214/ECP.v16-1688


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 $1_{x< B_t} dx$, $t>0$.


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Siva Athreya. Rongfeng Sun. "One-dimensional Voter Model Interface Revisited." Electron. Commun. Probab. 16 792 - 800, 2011.


Accepted: 7 December 2011; Published: 2011
First available in Project Euclid: 7 June 2016

zbMATH: 1243.60077
MathSciNet: MR2868600
Digital Object Identifier: 10.1214/ECP.v16-1688

Primary: 60K35
Secondary: 60F17, 82C22, 82C24


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