The contact process with fast voting

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 λ and voting events occur at rate θ. We are interested in the asymptotics as θ→∞ of the critical value λc(θ) for the existence of a nontrivial stationary distribution. In d≥3, λc(θ)→1/(2dρd) where ρd is the probability a d dimensional simple random walk does not return to its starting point.In d=2, λc(θ)/log(θ)→1/4π, while in d=1, λc(θ)/θ1/2 has lim inf≥1/2√ and lim sup<∞.The lower bound might be the right answer, but proving this, or even getting a reasonable upper bound, seems to be a difficult problem.

Duke Scholars

Author Richard Timothy Durrett Mathematics