M ar 2 00 6 THRESHOLD θ ≥ 2 CONTACT PROCESSES ON HOMOGENEOUS TREES Luiz

We study the threshold θ ≥ 2 contact process on a homogeneous tree Tb of degree κ = b + 1, with infection parameter λ ≥ 0 and started from a product measure with density p. The corresponding mean-field model displays a discontinuous transition at a critical point λMF c (κ, θ) and for λ ≥ λ MF c (κ, θ) it survives iff p ≥ p MF c (κ, θ, λ), where this critical density satisfies 0 < pMF c (κ, θ, λ) < 1, limλ→∞ p MF c (κ, θ, λ) = 0. For large b, we show that the process on Tb has a qualitatively similar behavior when λ is small, including the behavior at and close to the critical point λc(Tb, θ). In contrast, for large λ the behavior of the process on Tb is qualitatively distinct from that of the mean-field model in that the critical density has pc(Tb, θ,∞) := limλ→∞ pc(Tb, θ, λ) > 0. We also show that limb→∞ b λc(Tb, θ) = Φθ, where 1 < Φ2 < Φ3 < ..., limθ→∞ Φθ = ∞, and 0 < lim infb→∞ b θ/(θ−1) pc(Tb, θ,∞) ≤ lim supb→∞ b θ/(θ−1) pc(Tb, θ,∞) < ∞.