Electronic Journal of Probability

The contact process with fast voting

Rick Durrett, Thomas Liggett, and Yuan Zhang

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Consider a combination of the contact process and the voter model in which deaths occur at rate 1 per site, and across each edge between nearest neighbors births occur at rate $\lambda$ and voting events occur at rate $\theta$. We are interested in the asymptotics as $\theta \to\infty$ of the critical value $\lambda_c(\theta)$ for the existence of a nontrivial stationary distribution. In $d \ge 3$, $\lambda_c(\theta) \to 1/(2d\rho_d)$ where $\rho_d$ is the probability a $d$ dimensional simple random walk does not return to its starting point.In $d=2$, $\lambda_c(\theta)/\log(\theta) \to 1/4\pi$, while in $d=1$, $\lambda_c(\theta)/\theta^{1/2}$ has $\liminf \ge 1/\sqrt{2}$ and $\limsup < \infty$.The lower bound might be the right answer, but proving this, or even getting a reasonable upper bound, seems to be a difficult problem.

Article information

Electron. J. Probab., Volume 19 (2014), paper no. 28, 19 pp.

Accepted: 3 March 2014
First available in Project Euclid: 4 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]

contact process voter model block construction

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Durrett, Rick; Liggett, Thomas; Zhang, Yuan. The contact process with fast voting. Electron. J. Probab. 19 (2014), paper no. 28, 19 pp. doi:10.1214/EJP.v19-3021. https://projecteuclid.org/euclid.ejp/1465065670

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