One-dimensional voter model interface revisited

Athreya, Siva R. ; Sun, Rongfeng (2011) One-dimensional voter model interface revisited Electronic Communications in Probability, 16 (70). pp. 792-800. ISSN 1083-589X

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We consider the voter model on 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 γ-th moment for any γ >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 (Bt)t≥0 under diffusive scaling of space and time. This convergence fails when p(·) has an infinite γ-th moment for any γ < 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 (1x < Btdx)t ≥ 0.

Item Type:Article
Source:Copyright of this article belongs to Institute of Mathematical Statistics.
Keywords:Voter Model Interface; Measure-valued Process; Tightness
ID Code:100180
Deposited On:12 Feb 2018 12:16
Last Modified:12 Feb 2018 12:16

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