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→⫗ σ _N /logN = 1/γ₁(λ); 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. The arguments of Durrett and Liu and of Durrett and Schonmann hold for λ<λ _c1 and λ>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.