We continue the investigation of the behavior of the contact process on infinite connected graphs of bounded degree. Some questions left open by Salzano and Schonmann (1997) concerning the notions of complete convergence, partial convergence and the criterion $r = s$ are answered.
The continuity properties of the survival probability and the recurrence probability are studied. These order parameters are found to have a richer behavior than expected, with the possibility of the survival probability being discontinuous at or above the threshold for survival. A condition which guarantees the continuity of the survival probability above the survival point is introduced and exploited. The recurrence probability is shown to always be left-continuous above the recurrence point, and a necessary and sufficient condition for its right-continuity is introduced and exploited. It is shown that for homogeneous graphs the survival probability can only be discontinuous at the survival point, and the recurrence probability can only be discontinuous at the recurrence point.
For graphs which are obtained by joining a finite number of severed homogeneous trees by means of a finite number of vertices and edges, the survival point, the recurrence point and the discontinuity points of the survival and recurrence probabilities are located.
"The Second Lowest Extremal Invariant Measure of the Contact Process II." Ann. Probab. 27 (2) 845 - 875, April 1999. https://doi.org/10.1214/aop/1022677388