Phase Transition for the Contact Process in a Random Environment on Zd*Z+

We consider the basic contact process in a static random environment on the half space Zd*Z+, where the recovery rates are constants and the infection rates are proportional to a series of independent and identically distributed random variables. The environment can be seen as a u0027parameterizedu0027 version of Yao u0026 Chen(2012). We show that with probability one, the contact process at the critical value dies out. As a corollary, we can get that with probability one, the complete convergence theorem holds for all positive parameters. This is a generalization of the known results for the classical contact process in the half space case.