The contact process with fast voting

摘要

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 $heta$. We are interested in the asymptotics as $heta oinfty$ of the critical value $lambda_c(heta)$ for the existence of a nontrivial stationary distribution. In $d ge 3$, $lambda_c(heta) o 1/(2dho_d)$ where $ho_d$ is the probability a $d$ dimensional simple random walk does not return to its starting point.In $d=2$, $lambda_c(heta)/log(heta) o 1/4pi$, while in $d=1$, $lambda_c(heta)/heta^{1/2}$ has $liminf ge 1/sqrt,$ 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.