The Annals of Applied Probability

Equilibrium interfaces of biased voter models

Rongfeng Sun, Jan M. Swart, and Jinjiong Yu

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A one-dimensional interacting particle system is said to exhibit interface tightness if starting in an initial condition describing the interface between two constant configurations of different types, the process modulo translations is positive recurrent. In a biological setting, this describes two populations that do not mix, and it is believed to be a common phenomenon in one-dimensional particle systems. Interface tightness has been proved for voter models satisfying a finite second moment condition on the rates. We extend this to biased voter models. Furthermore, we show that the distribution of the equilibrium interface for the biased voter model converges to that of the voter model when the bias parameter tends to zero. A key ingredient is an identity for the expected number of boundaries in the equilibrium voter model interface, which is of independent interest.

Article information

Ann. Appl. Probab., Volume 29, Number 4 (2019), 2556-2593.

Received: April 2018
Revised: October 2018
First available in Project Euclid: 23 July 2019

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Mathematical Reviews number (MathSciNet)

Primary: 82C22: Interacting particle systems [See also 60K35] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82C24: Interface problems; diffusion-limited aggregation 82C41: Dynamics of random walks, random surfaces, lattice animals, etc. [See also 60G50] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Biased voter model interface tightness branching and coalescing random walks


Sun, Rongfeng; Swart, Jan M.; Yu, Jinjiong. Equilibrium interfaces of biased voter models. Ann. Appl. Probab. 29 (2019), no. 4, 2556--2593. doi:10.1214/19-AAP1461.

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