Critical value for the contact process with random recovery rates and edge weights on regular tree

In this paper we are concerned with contact processes with random recovery rates and edge weights on rooted regular trees TN. Let ρ and ξ be two nonnegative random variables such that P(ϵ≤ξ<+∞,ρ≤M)=1 for some ϵ,M>0. For each vertex x on TN, ξ(x) is an independent copy of ξ while for each edge e on TN, ρ(e) is an independent copy of ρ. An infected vertex x becomes healthy at rate ξ(x) while an infected vertex y infects an healthy neighbor z at rate proportional to ρ(y,z). For this model, we prove that the critical value under the annealed measure approximately equals (NEρE1ξ)−1 as N grows to infinity. Furthermore, we show that the critical value under the quenched measure equals that under the annealed measure when the cluster containing the root formed with edges with positive weights is infinite.