# zbMATH — the first resource for mathematics

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
Full Text: