Contact processes with random vertex weights on oriented lattices

Abstract

In this paper we are concerned with contact processes with random vertex weights on oriented lattices. In our model, we assume that each vertex x of Z^d takes i. i. d. positive random value \rho(x). Vertex y infects vertex x at rate proportional to \rho(x)\rho(y) when and only when there is an oriented edge from y to x. We give the definition of the critical value \lambda_c of infection rate under the annealed measure and show that \lambda_c=[1+o(1)]/(dE\rho^2) as d grows to infinity. Classic contact processes on oriented lattices and contact processes on clusters of oriented site percolation are two special cases of our model.