MULTIPLE TRANSITION POINTS FOR THE CONTACT PROCESS ON THE BINARY-TREE

The contact process on Z(d) is known to have only two fundamental type s of behavior: survival and extinction. Recently Pemantle discovered t hat the phase structure for the contact process on a tree can be more complex. There are three possible types of behavior: strong survival, weak. survival and extinction. He proved that all three occur on homog eneous trees in which each vertex has d + 1 neighbors, provided that d greater than or equal to 3, but he left, open the case d = 2. Since d = 1 corresponds to Z(1), in which weak survival does not occur, d = 2 is the boundary case. In this paper, we complete this picture, by sho wing that weak survival does occur on the binary tree for appropriate parameter values. In doing so, we extend and develop techniques for ob taining upper and lower bounds for the critical values associated with strong and weak survival of the contact process on more general graph s.