On the Critical Behavior at the Lower Phase Transition of the Contact Process by Michael Aizenman

We present general results for the contact process by a method which applies to all transitive graphs of bounded degree, including graphs of exponential growth. The model’s infection rates are varied though a common control parameter, for which two natural transition points are defined as: i. λT , the value up to which the infection dies out exponentially fast if introduced at a single site, and ii. λH , the threshold for the existence of an invariant measure with a non-vanishing density of infected sites. It is shown here that for all transitive graphs the two thresholds coincide. The method, which proceeds through partial differential inequalities for the infection density, yields also generally valid bounds on two related critical exponents. The work extends existing results whose derivations were restricted to either the discrete-time versions of the contact process or to graphs with subexponential growth.