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The contact process with fast voting. (English) Zbl 1291.60203

Summary: 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 \(d \geq 3, \lambda_c(\theta) \to 1/(2d\rho_d)\) 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 \(\liminf \geq 1/\sqrt{2}\) 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.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
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