The Contact Process with Fast Voting
Richard Durrett, Thomas M. Liggett (UCLA), and Yuan Zhang
Abstract.
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. Let ρd is the probability a d
dimensional simple random walk does not return to its starting point.
- In dimensions d ≥ 3, λc(θ) → 1/(2dρd)
- In d = 2, λc(θ)/log(θ) → 1/4π.
- In d = 1,
λc(θ)/θ1/2 has liminf ≥ 2-1/2
and limsup < ∞.
The lower bound might be the right answer, but proving this, or even getting a reasonable upper bound, seems to be a difficult problem.
Preprint
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