The asymmetric contact process on a finite set

Abstract

The contact process onZ has one phase transition; let λ c be the critical value at which the transition occurs. Let σ N be the extinction time of the contact process on {0,...,N}. Durrett and Liu (1988), Durrett and Schonmann (1988), and Durrett, Schonmann, and Tanaka (1989) have respectively proved that the subcritical, supercritical, and critical phases can be characterized using a large finite system (instead ofZ) in the following way. There are constants γ1(λ) and γ2(λ) such that if λ λ c , lim N→⫗ log σ N /N = γ2(λ); if λ=λ c , lim N→⫗ σ N /N=∞ and lim N→⫗ σ N /N 4=0 in probability. In this paper we consider the asymmetric contact process onZ when it has two distinct critical values λ c1 λ c2. We show that for λ∈[λ c1<λ c2), lim N→⫗ σ N /N=-1/α, (where α i is an edge speed) and for λ=λ c2, lim N→⫗ logσ N /logN=2 in probability.